I understand that the common mistake is to assume that the odds of landing 5 blacks in a row is no worse than 4 blacks and then 1 red BUT...can't one assume that if you played red the entire game you would win approximately 40-60% of the time nearly every game where that was your strategy? (Not saying this is a good strategy!)
Basically, I sit down at a black jack table and use the board to influence my gambling...is it just lucky that it always helps me win (I'll see its been black 7 out of the last ten and I will play red and usually win)?

I administrate a small gacha community myself and feel like these are information that need to be shared. I'm not telling you that you shouldn't or can't spend money on Genshin Impact, but the least you can do is being aware of these things.
Whether you are already caught in the trap or not, simply knowing about these tricks, traps and fallacies will help you. Games use them to lure you in, keep you engaged and tempt you to spend money. I'll try to keep these points as short and condensed as I can.

0. Am I genuinely having fun?

Since this one is not really a trap and thus something of an outlier I marked it as 0., but it's a great question to keep in mind: "Do I still have genuinely fun playing this game, or does it feel like a chore? Is this worth my limited time?" You should regularly ask yourself that question.

1. The monthly pass (Blessing of the Welkin Moon) has a great price-performance ratio and is cheap so it can't hurt... can it?

The monthly passes in games are more dangerous than you think:

• You get part of your purchase only on a daily basis to make you log in every day to not waste the gems you bought. This is to condition you to make the game your daily routine.
• Using a cheap thing you can buy makes you set up a payment option and makes you accustomed with the process of buying. This reduces the resistance in case you have the urge for an impulse purchase.
• Just because the price-performance ratio is better than the other available options doesn't make $4.99/5,50€ automatically worth it. You're still spending money on virtual things.

2. It's free to play and I enjoy(ed) it, so spending some money can't hurt.

While technically not wrong, be aware of the consequences this can have. Similar to the points above it might also be a trigger to incentivize for further purchases. Especially considering the next point:

3. Guilt-tripping the player into playing after they spent money

The game doesn't even have to do much for this since it usually happens by itself. Once you paid money you feel like you now have to play, even when you're not even really having fun.

4. The daily routine of work and chores

Just like with the monthly pass, by rewarding daily engagement with the game (daily quests/battle pass/rotating dungeons) you get conditioned to make the game part of your daily schedule/life. Once logging in and playing becomes a habit it can be difficult to stop, even if you don't really enjoy it and are literally just doing in-game work/chores.

5. The first time is free - veiling the distribution of currencies/gems

Also known as the honeymoon phase. You get a lot of free stuff in the beginning, tempting you to roll and get a taste of what rolling feels like - especially if you're lucky and get a rare characteitem. By slowly decreasing the distribution of these currencies you feel tempted to spend money to get more of that initial high you had. This hits especially bad when new content is shown/announced while you are out of gems.

6. Overwhelming the player with a lot of different currencies/gems

Genesis Crystals, Primogems, Intertwined Fate, Acquaint Fate, Starglitter, Stardust, there are so many (more or less) premium currencies that it's almost confusing - and not by accident.

• Multiple currencies are used to make the player lose track of the different values. Buying something should make the player also think they got "a lot of different things" when it's just real money in different forms.
• By sneaking in multiple currencies, the value of money is lost along the way. Example: Real money -> Primogems -> Intertwined Fate - This blurs the link between money and Intertwined Fate.

7. The carrot and the stick - New content and changing the meta

"Now that I have the best/meta character I will never need to spend money again." is sadly a trap a lot of people fall for, but what is currently considered "the best" will definitely change.
This is an investment into something that will not last, and even worse is that since you already spent a lot of money in the past you feel pressured to get the new "best" thing to not lose your place on top of the player-base.

8. "It's okay, I only spent $___, some other people spent waaay more."

If you're telling yourself something like this then you already walked further into the gacha trap than you might think. Denial in the form of rationalization is a crucial and dangerous sign for gambling addiction.

9. Gambler's Fallacy and Sunk Cost Fallacy - A dangerous combination

Well known classics, but I still felt like including them.
Gambler's Fallacy: "Gambler's fallacy refers to the erroneous thinking that a certain event is more or less likely, given a previous series of events."
Sunk Cost Fallacy: "Individuals commit the sunk cost fallacy when they continue a behavior or endeavor as a result of previously invested resources (time, money or effort)."
While already dangerous on their own, in combination these two reach a whole different level. Even with the pity system, just because you were unlucky a lot doesn't mean that you are guaranteed to become more lucky in the future. This in combination with spending a lot of money without getting what you want can tempt you to spend even more.
"I already spent $200 and didn't get what I want. If I stop now those $200 will be wasted!"
This is a highly dangerous train of thought!

10. The Sunk Cost Fallacy strikes again

You don't have to spend actual money to fall for this fallacy. If the only thing that keeps you playing is the amount of time you already spent, the items you own and the characters you unlocked, then maybe you should reevaluate your decision. "If it's not fun, why bother?"
 
 
There are a lot more techniques that are used to trick people, from flashy pull animations to making you aware and jealous of other people's characters/pulls (Why do you think phone players are incentivized to take a screenshot of their pulls and share it?), but I wanted to keep it to the most common/obvious ones.
Edit: If you're still curious and want to know even more tricks and traps, Dokuganryu linked a great and informative video. Definitely worth a watch if you have the time.
 
I genuinely hope everybody is enjoying Genshin Impact, but keep in mind how this company makes their money.

Hey all, so from just perusing some of the top video makers on youtube for this game and the servers I play on it's come to my attention a lot of players don't actually know/understand the odds of getting the character you want from banners. So I've done this sporadically on a few of PayneBlades videos (shoutout to him for putting up with my shit) but I figured I'd make a post here and hopefully shed some light to some people on the actual numbers behind summoning. The goal of this is so that people can better understand their odds and make smarter decisions on when to summon. So there aren't so many "I can't believe I got shafted, this is BS!" comments all the time.
​
I'm going to show the following points from a statistical POV:

• How many summons do you need to pull the banner UR.
  • How many shino coins you need to do this.
• How many summons do you need to pull one of the EF Banner URs
  • How many shino coins you need to do this.
• What are your odds of pulling the Banner UR when you get a UR?
• How to figure out your chance of getting the banner UR in X shino coins

​
There is a TL;DR at the bottom that summarizes the main stats, so feel free to skip to that if you don't want to read through this math.
​
NOTE: There are currently no normal double UR Banners so I don't know their rates and thus cannot calculate that. I can update this post when a normal double UR banner comes.
​
How many summons do you need statistically to get the banner UR:
Total odds of getting the banner character = odds of getting 3* banner + odds of getting 4* banner
= .047% + .660% = .707% (This means on each summon you have a .707% chance of pulling the banner UR in 3* or 4* form in other words you will succeed .707 times out of 100. Do note I'm not saying the chance of pulling the 4* is .707% just that the chance of either event happening sums to .707%)
.707% / 100 = .00707 (Divide by 100 to remove the %)
1 / .00707 = 141.442 ~ 142 (Can't have half a summon)
142 - 1 = 141 (141 to 1 is the odds ratio. This means for every one success you will have 141 failures.)
Now that we have the total number (142) let's see how many shino coins that is:
142 * 250 = 35500 shino coins
This means statistically you'll need 35500 shino coins to get the feature unit on the banner. Obviously there is a .707% chance you can get the banner unit on any single summon. So some of you may summon the banner UR from their first pull, but the chance of not summoning the banner unit each summon is:
1 - .00707 = .99293 * 100 = 99.293%
So every time you summon you have 99.293% chance of NOT summoning the banner unit.
​
How many Summons do you need to get the banner UR on an EF banner:
Total odds of getting one of the URs = odds of 3* + odds of 4*
= .1% + 1.4% = 1.5% ( This is 2.12x compared to a standard banner. Almost 4.5 times as much if we compared getting either)
1.5 / 100 = .015
1 / .015 = 66.667 ~ 67 (can't have partial summon)
67 - 1 = 66 ( 66 to 1 is the odds ratio. Every success will contain 66 failures)
67 * 250 = 16750 shino coins
This is 47% (less than half) the cost of a normal banner. So as you can see it's a very smart idea to wait for the EF banner if you aren't a whale. As I said above I don't know the rates for a double UR banner but I would assume they're slightly worse than a normal UR banner because more units in the pool is usually a reduced rate. Even if that assumption is incorrect with a double UR banner you could likely purchase one (maybe both? I'm not sure how the shop works) from the shinobi exchange and then get the shards from FG/Abyss and then summon the unit on the corresponding EF banner.
​
What are your odds of pulling the Banner UR when you get a UR?
In EF banners it's 50% (Super easy, only two units both have equal rates so it's split right down the middle)
Normal Banner:
Total chance of pulling a UR (4* or 3*) is 3%, as we saw above the chance of pulling the banner UR is 0.707%
.707 / 3 = 0.235667 * 100 = 23.56%
So whenever you pull a UR there is a 23.56% chance it is the banner UR. Compared to EF where it's 50% you can again see EF banners give just over twice the chance of getting the UR you want.
​
How to figure out your chance of getting the banner UR in X shino coins
So this is (in my opinion) the best one. This is a way for anyone to see what their odds of getting the banner UR is with the amount of Shino coins they currently have.
Chance of getting banner UR at least once = 1 - chance of not getting banner UR
This makes sense because in percentages things need to add up to 100 (or one if you want to look at just the raw number and not the %). If if you know the odds of some event happening then the compliment (in other words the opposite of that event) have the remaining odds needed to sum up to 100%.
For example if you're watching the Super Bowl and the announcer says "The Chiefs have a 98% chance to win the game right now" it means the chance of them losing the game is 2% (1 - .98 = .02, .02 * 100 = 2%)
So we need to calculate the chance of not getting a banner UR in X (your number of summons) tries. This is actually easier than it sounds because the chance of not getting a banner UR is the SAME every single time you summon (another word for this is independent events. Meaning whatever happened on the previous try has no effect on the current try)
It just changes out equation to:
chance of getting banner UR at least once = 1 - (chance of not getting banner UR)^{number of summons}
**chance of getting banner UR at least once = 1 - (.9923)**^{number of summons}
Now that the equation is all laid out here are the steps:

1. Calculate number of summons you have = number of shino coins / 250 (round DOWN. Can't have a partial summon)
2. Calculate (.9923)^{number of summons} (As we saw above chance of not getting banner UR is .9923)
3. Calculate chance of getting banner UR = 1 - your answer to #2
4. (optional) multiply answer to #3 by 100 to get the % chance

An example:

1. I have 10,000 shino coins. 10,000 / 250 = 40
2. (.9923)⁴⁰ = .734039 (You can google .9923^40 to have it calculate the answer for you)
3. 1 - .734039 = .26961
4. .26961 * 100 = 26.961% chance of pulling the banner UR.

Here are the equations needed to calculate getting 1 of the EF characters and getting either of the EF characters:

• chance of getting one EF UR at least once = 1 - (.985)^{number of summons}
• chance of getting either EF UR at least once = 1 - (.97)^{number of summons}

​
TL;DR:
It's very low odds to summon the banner unit on a normal banner. If you're not a Whale you should NOT summon on every banner. You should save resources as much as possible to give yourself the best chance to get the banner unit. EF Banners are a way better chance EVEN if one of the units is not desirable.
​
TL;DR Stats:
(All include the 4* or 3* chance added together)

• Normal Banner stats
  • You have a .707% to summon the banner UR on a normal banner summons each time you summon. (This is the chance of the 3* or 4* summed.)
  • If you were to pull the Banner UR there is a 6% chance it's the 4* version.
  • You have a 99.293% chance to NOT summon the banner UR on a normal banner summons each time you summon.
  • The odds ratio for summoning the banner UR is 141 to 1 (Meaning for each success you will have 141 failures)
  • 142 summons is 35500 shino coins. Meaning you have to spend that much before it becomes a "shaft" anything under that and you've not met the odds ratio so it may be unlucky compared to others but it's not uncommon from a straight odds perspective.
• EF Banner Stats
  • You have a 1.5% chance to summon one of the banner URs on an EF banner.
  • The odds ratio for summoning one of the banner URs on an EF banner is 66 to 1 (meaning for each success you will have 66 failures)
  • 67 summons is 16750 shino coins. Meaning you have to spend that much before the banner has shafted you. (Not here I didn't calculate summoning both meaning if you summoned the unit you didn't want you got a coin flip the wrong way.
• Odds of pulling Banner UR when you pull a UR
  • For EF it's 50% chance. When you pull a UR there's a coin flip between which it will be.
  • Normal Banners you have a 23.56% chance to get the banner unit when you summon a UR
• What is my % chance of getting the banner unit when summoning with X shino coins
  • Shino coins / 250 = number of summons
  • chance of getting banner UR at least once = 1 - (.9923)^{number of summons}
  • chance of getting one EF UR at least once = 1 - (.985)^{number of summons}
  • chance of getting either EF UR at least once = 1 - (.97)^{number of summons}

​
Common misconceptions:

• "But pedantic programer I got the banner unit in one single. So this math doesn't apply to me"
  • No it still does. You just achieved that .707% chance on your first try. If 30,000 people all summon once on a banner there is a pretty much guaranteed chance (.9999999 to like the 100th place) that at least one of them will get the banner unit on their first summon.
• "But pp I didn't get the banner unit in 75k coins so this isn't true"
  • This is the exact opposite of the previous answer. It's RNG so while your chance of getting the banner unit with 75k coins is high (90.2%) there's still the chance you don't get it. You got shafted. That sucks pray to RNGesus
• "I spent 40k, I should get him/her anytime now right?"
  • Wrong, this is pretty much the definition of gamblers fallacy and your previous 40k summons have absolutely 0 effect on the next 40k summons. A lot of gacha games have implemented spark systems to alleviate this. So after X amount of coins spent you will be guaranteed the banner unit. I hope tribes implements one but until then there is not magic cure to guarantee you'll get the unit.

I was playing a video game (Overwatch) the other day and my friend said 'it doesn't seem like you're having fun, you just like winning.' I responded: 'it's not even that I like winning, I just hate losing.' Moreso I hate feeling like I'm going to lose and can't do anything about it. I hate feeling powerless. It makes me want to become a machine gun turret and just mow down everyone in order to beat the enemy team (Bastion). Using that 'sledgehammer' method to overpower the other team can be very satisfying, but it's only one tactic - there is a limit to how much force alone can do in a situation. Imagine playing chess when all you know is going for the queen at the expense of everything else - it just makes you lose faster, unless you somehow get lucky. So I started to make this connection -> it was triggering a 'fight response.' A black and white desire to overpower the enemy which was making me feel so powerless. So much of how the game in Overwatch goes at the lower levels is up to luck. I felt like a gambler who thought I could control luck and just kept going and going until all I had was all gone.
So I was thinking, how can I recognize these trauma responses? Emotionally neglected people can struggle to recognize, connect, and process these feelings/states - perhaps thinking these trauma responses are the only reality instead of a triggered response affecting perception/shutting down cognitive functioning. I'm mainly wrote this to flesh it out for myself. I tried to put a proposed mapping from my experience below, and I would appreciate any thoughts on this!

1. Fight response: feels like desperately trying to control a situation with blunt force/powerepetition/arguments, etc. Often caused by feeling powerless. Possibly connected to my general anxiety disorder. The cognitive fallacy is that I think I can control more than I can in the situation - perhaps based on previous experiences from my memory where brute force seemingly worked, but was probably due to dumb luck (effort/force can help up to a point in some situations, but adaptability/situational awareness is more important).
   
2. Flight response: I can't focus on anything and get impulsively task switch. Many poducts feed off this attention (eg. opening reddit, closing it, then opening it up again without realizing it.). I also sometimes Impulsively snack or make and change plans as part of this. Deep down I'm running from my feeling of overwhelm and exhausted (gets worse when I've slept less). I'm just spinning my wheels but getting nowhere. This is probably connected to ADHD for me. Cognitive fallacy is that I'm still making progress when really I'm not and I need to let go and just take a break (often when I take a break to lie down in this state I fall asleep).
   
3. Freeze response: feeling a deep shame and wanting to disappear completely (sometimes I imagine the sunken place place from Get Out). This one was the hardest for me to reach in therapy, I think because it overlaps with dissociating/memory/magical thinking/imagination. It's internal panicking taking over external functioning - my circulation seems to really slow down and I get really cold. I wrote another post about this experience late at night here. Honestly the cognitive fallacy here for me is that this state doesn't exist/isn't real - when it definitely is.
   
4. Fawn response: worshipping people who are powerful (intelligent, skilled, attractive, rich, famous, etc) - or any hero worship. This one includes the people/situation I'm in from a social lens and can come up as social anxiety for me when I'm feeling unsure about what to do in a social situation. The consequence is that I drop my sense of self and taking on the identity/value of the group - I become a chameleon. The hard part afterwards is figuring out who I am as distinct from who the situation/other's voices in my head are. Perhaps this is related to some attachment trauma for me. Cognitive fallacy is that if I side with the bully I will be spared, when really thats not necessarily true - also, no human is perfect. I can end up brainwashing myself (eg. Stockholm syndrome) and/or overly obsessing about the person. This one actually works often at scale (eg Trump worship in America the past few years), but the downside is that it hurts the overall society to worship idols as infallible. I'm the least familiar with this one (fawn), but I also read a common one is to try to help/'fix' friends as a way to avoid helping yourself.
   

Other notes/connections:

• Games can be helpful to practice engaging your mind differently. For example in Overwatch - I started focusing on aim over other skills. So when I play I focus on precision, not force. I usually do this as part of warming up to play more competitively with others - and I realized in retrospect the reason it helped was less so mechanics than just engaging my mind differently -> getting me out of 'brute force' mindset if things start to go badly and focus more on the task at hand.
  
• Practicing stuff for me (eg sports, video games, etc) is more about processing past anxiety, so I am more present for the future. Not making myself invulnerable, but allowing myself to accept my vulnerability more gracefully. No one can control the future, but being more present and aware is almost always better (luck is probably best...).
  
• A note from my individual therapy when I asked 'how do you tell if you're the asshole?' -> You feel like a victim in the moment. You'll feel justified to do whatever because of the pain you're in. For example, Trump saying 'you stole the election!!' So they can do whatever they want in response via a blank check.
  

Speedrunning is a 👏👌 hobby in ☝ which people 😡👨 compete to complete 🚫 a 😱👌 video ♀ game 🏈🎲 as 🛠🍑 quickly 🆘⏰ as 😎 possible. 🔝 This paper 🤓 concerns speedruns of 💦 Minecraft: ⛏🚨 Java Edition, and, 👎😖 in 🚫 particular, speedruns of 🚨💦 the category known 💫 as 🍑💑 "Any% 🤣 Random 🎲🔀 Seed 👨🌾 Glitchless" (RSG) performed on 🔛 version 👧 1.16. 👸🏻 A 🅱 brief summary of 💦 the 👏🔍 relevant mechanics 💰 and speedrun strategies follows 🚗♀ for 👀 the unfamiliar reader. The 🗣 final 😪 boss 👨👨 of 👼💦 Minecraft ⛏ is located in 👄📥 an alternate dimension known as 🚫 The 🏕💦 End, 😣🔚 which 🙌😩 can be 🐝🐝 accessed using 🤳🏻 End ✋💯 Portals. An end 🖕✋ portal consists of 💰☹ twelve End Portal Frame blocks, 🌆🌇 a ⛄☝ random 🔀🔀 number 🎦 (usually 10-12) 🅾 of 💦 which 👏 must be filled 🔋😩 with 😍👫 an ☺ Eye 🤕 of 💦 Ender in 🚭👇 order to activate the portal. Thus, 👌🕵 the 🌞 runner is ☎👅 required ✅📋 to 👊💦 have ✊💪 up to twelve eyes 👀 of 💦 ender when 💕 they arrive at the 🤡🏻 portal to 💰🗣 be 😡🐝 able 💪💪 to 💦 enter The End and 👏👏 complete the game. 🏈🎮 In 📥 1.16, the ♀ only way 😓 to obtain an 👹 eye 👁 of ender is ❤♂ by crafting it, 🏃 which requires one Ender Pearl and one ☝👏 Blaze Powder. Ender pearls 🍬 can be 🐝🐝 obtained in several ⛓ ways, 💯😉 but 🏼🍑 the fastest is 🔥👌 to 💦👊 use a 🔥🐝 mechanic known 😝💫 as Bartering. In 🛌 a 🅱 barter, the 🔛 player 👨🎮 exchanges a 🏻👬 Gold 🤓🔦 Ingot with a 👍 Piglin (a humanoid creature 🐙 in the Nether dimension) for 👻 a randomly chosen item or 💰😫 group 🅱🅱 of 🚋👏 items. 🛡🛡 For 🏻🙃 each 👏 barter, there is 👅 about 🏫 a 5% chance ♂🙅 (in 📥👏 1.16.1) 🌱🚫 the piglin will 🤔 give 🏿 the 👏🕡 player ♀💰 ender pearls. 🍬🍬 Blaze powder is crafted out of 🏫 Blaze Rods, which are 😡💯 dropped 〽 by Blazes—a hostile mob. Upon being ❌ killed, each blaze has a 50% chance 😨 of 👨🎖 dropping one blaze rod. 🍆🍆 The 👏💛 main focus 😩 during 🚣🏃 the 🦉 beginning ➡😍 of 💦🔥 a 🔫👇 1.16 ⛈ RSG speedrun is to ✌💰 obtain (hopefully) 👏👏 12 🤓😣 eyes 👀 of 🍳👩 ender as 🙇 quickly ⏰ as 🏿 possible, by 🌈😈 bartering with 🆕👉 piglins and 🚕 killing blazes. These two 💏💏 parts of 😩💦 the 🏻🌊 speedrun route are 🔢🙏 the 👏💦 primary concern 😕 of this paper. 2 😂 Motivation ☑ Members 👨👨 of 😏 the 🏻 Minecraft ⛏ speedrunning communitya reviewed six consecutive livestreams of 😣 1.16 📣🔙 RSG speedrun attempts by 😈 Dreamb from 👩🛣 early October 2020. The 👶 data 💰💰 collected show 📺 that 🍆👀 42 of the 💬 262 piglin barters performed 🎭💃 throughout these streams yielded ender pearls, 🍬 and 💰🙋 211 of 🔧 the ⏰👀 305 killed blazes dropped blaze rods. These results 🔢 are 💇 exceptional, as 😠 the 😍🏽 expected proportion of 💦💦 ender pearl barters and 😵 blaze rod 🍆🍆 drops 💦😲 is 🙀 much, 😣👎 much lower. 🤓 An 👹💉 initially compelling counterclaim is 🍆😀 that 💦 top-level 🔼 RSG runners must 🙅 get 🍑 reasonably good 🏼💘 luck in 👌🏼 order to 💦💦 get a ♀👌 new 🎉🤤 personal best 👳👌 time 🕐⏰ in 👻😈 the 👨 first 🥇😂 place, 🏆🤤 so, while ♀👶 it 💯➡ is 🙈 surprising to 💦💀 see 👁 such 💦 an ➕ unlikely event, it 💯🦊 is perhaps 😍🏻 not 😥😅 unexpected. However, 🖐💯 upon 👦 further research, Dream’s 💭 observed drop 🏻👇 rates 💦 are substantially greater 👅👅 than 😽 those 👞👞 of 💦😰 other top-level 🔼 RSG runners—including, Illumina, Benex, Sizzler, and Vadikus. If ☔☔ nothing ♀🔫 else, 👴♀ the 😱🌈 drop 👇⤵ rates from 💋 Dream’s 💭💭 streams are 💓 so exceptional that 💯 they 🏽 ought to be 🐝 analyzed for 💰 the sake of it, 🔥 regardless of 💦 whether ☂☂ or not 🚫🚫 any one 😍 individual believes they 🏾 happened 🤔🤔 legitimately. aThe data 💾 were 😫😫 originally 🔙🔙 reported ♂ by MinecrAvenger and danhem9. bhttps://www.twitch.tv/dreamwastaken 3 3 😘🤔 Objectivity 🤖 The 🙀🐺 reader should 👫 note that the 🏻 authors of 💦 this 😎📍 document are solely 👞👞 motivated 🏿 by 😈 the presence 🙇 of exceptional empirical data, 📉💰 and 👏👎 that 😐🎢 any 👏 runner—regardless of 🏻😍 popularity, following, or 👉💰 skill— 😤 observed experiencing such unlikely events would 😏😎 be 😤 held to ✌💦 the 😼👌 same 😯 level 🔻🆙 of scrutiny. The 👊🅿 reader should 💦👍 also 🙇 note that 🤔🚟 the data 📉💰 presented are 🏻😩 extensively corrected for 🤔🎅 the 😫📚 existence 👌 of 🔴 any 🈸 bias. It 🔫😞 would 🍆😎 lack rigor and 💰💦 integrity for 💕😩 the 🗯 conclusions made 👆 in this 😏 report to 💦 substantiate the moderation 💯💯 team’s 🐒 decision if they 👥 were 👧👶 merely based 👌 on 🚟🔛 a surface-level 🍑🌎 analysis of the data. 📊 Indeed, these 🚱 corrections inherently skew the analysis in Dream’s 💭 favor. We 😂👦 aim to calculate not 🚫🖐 the 🍩💋 exact 👌👌 probability that 🏻👉 this 👈 streak 🔥 of 💯 luck 😄 occurred if Dream 💭 is 🔁 innocent, 😇😳 but 🍑 an upper bound 🤐🤐 on 🔛🔛 the probability; that is, 💦 we will 👊👻 arrive at 🚓😁 a value which 😡👏 we are certain is 😳🔮 greater 👅👅 than 👉 the 🅱💦 true probability. The 👦 goal 💦⚽ of 🏧👉 this ☝👈 document is to 💦⏸ present the unbiased, rigorous statistical analysis of the 👏🏼 data, 💰📉 as 😱🖕 well as an 🐎 analysis of the ♀🚨 Minecraft ⛏🚨 source 😔🏞 code, 😲😲 to conclusively determine whether or 💰➕ not 🚫 such an 🍎 event could be 👌😳 observed legitimately. 4 👌 Part 🏻 II 👩👩 Data The 📈💲 raw data 💾 (and its sources) 👉👉 from 🤤 which the 💰 following graphs were 👶 derived 🔜 can 💦💦 be 👏 found in 🍆 Appendix A. 🏿🅰 4 🙇 Piglin Bartering Figure 1: 🏫 Dream’s pearl barters, charted alongside various comparisons. The ✊ 99.9th 🤑 percentile line 🚫💨 represents one-in-a-thousand 💦 luck 😰🍀 (calculated 🚜🚜 using 📤📤 a 🥇➡ normal 🖖 approximation), which 🎓🙌 is already quite 💬🅰 unlikely—if not ♀ necessarily proof 💯📊 of anything. 😫 5 🍆 5 🍆🦐 Blaze Rod Drops 😲😲 Figure 2: ➡ The same for 🍆😱 blaze rod 🍆 drops. 💦 Part 👏 III Analysis 6 ❗ Methodology What ❓ follows 🏃🚗 is a thorough description 👿👿 of 😱 every aspect of our 🚟🅱 investigation in 👮 an 🍑 accessible manner. 🚁🚁 We 💰🔝 will begin 📦 with 👏👏 an 💸😯 introduction to the 👏👏 binomial distribution, and follow 👣 with 👏 adjustments 💰💰 to 🙅💊 account 💳 for 🔜🍆 sampling bias and 💰 other biases lowering the 👶 accuracy of 💦 the binomial distribution. Finally, 🙌 we 😂 will analyze Minecraft’s ⛏ code 😤 to justify the assumptions made 🙌💯 in our 👵💰 statistical model. 👄 To 🏻 strengthen 💪 our analysis to 💦🙏 the 💦👑 skeptical reader, we now 🍑 preemptively address 📪📫 expected 🤰🙄 criticisms and 👅👏 questions. Why 😩❓ are 🏻👀 you not 🏻 analyzing all of 😤💦 Dream’s runs? 💰💰 Doesn’t that introduce sampling bias? Yes. There is clearly 🤓😱 sampling bias in 👏 the 😦👌 data 📉 set, but 🤤🤚 its ⚜ presence 🏽 does 👏 not 🏼 invalidate our analysis. Sampling bias is a common 😍 problem in 👉👏 real-world ✨ statistical analysis, so if 🏿 it were 👩 impossible 🙇 to 🗣 account 💳 for, ⌛ then 🏿 every analysis of empirical data 💰 would be 🐝🍆 biased and 🏳👏 useless. ♀⏳ Consider flipping a 👌 coin 100 💯💯 times 🕛 and getting heads 🙉 50 👌 of 🔌 those times 💦 (a 🤣🤧 mostly 💁🙋 unremarkable result). Within 😱 those 100 coin flips, however, imagine 🤔 that 👏 20 of 👩 the 50 ⏳ heads 💤🙉 occurred back-toback somewhere within 😜 the 🙀➡ population. 👥 Despite 🙅♂ the 🐐☺ proportion overall being 😩 uninteresting, we 6 still would 😜 not expect 🤗🤗 20 📊 consecutive heads anywhere. Obviously, 🎳 choosing to 🔍 investigate the 👏 20 🔳🔳 heads 🙈🙊 introduces sampling bias—since we 👩 chose to ♂🙌 look 👀 at 🏠 those 😘 20 🎉🔳 flips because 💁😡 they were lucky, 🍀 we 👉💰 took a 👏 biased sample. However, 🖐🖐 we can instead discuss the probability that ☃👨 20 🆗🆗 or 💁🚫 more back-to-back 😡 heads occur 👻 at 👒😩 any 🚸 point in the 🔚 100 💯 flips. We 🏃👦 can use 👏 that 👆 value 💵💵 to 😠 place an 💰 upper bound 🤐🤐 on the probability that the 👏⛪ sample we chose could 🔒 possibly have 😣✅ been 👦 found 🚫👁 with 😣 a fair 👒 coin, regardless 🤷🤷 of how biased a 💰📖 method was 🅰 used to 🏽😩 choose the 🐆 sample. It’s 💦🏹 also 🙇🅱 worth 💵💸 noting that the choice to only 🕦 consider Dream’s 💭💭 most 💯😃 recent streak of 😳 1.16 🤜 streams is 👏 the 🔝 least arbitrary distinction we could 👌🚫 have 🙌👏 made. 👆 The 🖱🏋 metaphor of 🅱👶 "cherry-picking" 🍒 usually brings to ➡✌ mind 🤔 choosing from ➡ a wide ➡ number of 👩 options, but there ✔👇 were 🍑 at 🍆 most 💯 a 👌👌 small 😂🏼 handful of 🌹 options meaningfully equivalent to 💦 analyzing every 🏦 stream 💦 since 💦 Dream’s return to 💦♀ public 😪⛱ streaming. Note the 💌 importance of 😤💦 the 🔍 restriction that we ♀ must 🙅💰 analyze the 🍆😫 entire 🌲 six streams as a 💡👌 whole; 💰 true cherry-picking would 💭🌨 specifically select ❇ individual 🥖 barters to 💦💦 support 👍♀ a 👏 desired conclusion. How 🐼 do 👀 we 🅰 know this investigation isn’t biased? Concerns about 🤔 the 🏛👑 impartiality of the authors of 💦😊 this 😞 paper 🤓 have 👏😏 been raised 👧👧 in discussion about 🌈 the investigation. We 👨👩 do not think 💭 this 😞 is 🤔👐 a 🍾 significant issue; 🙅🏾 we have 😒 made 👆 an 😤 effort to be as 🍑 fair 👒 to 👸 Dream 💭💭 and ♂💰 thorough as 🖕🏿 possible in our investigation. Regardless, 🤷 it 💡💨 is a 👬 concern 😕😕 worth addressing. This 👈♂ paper 🤓 has 👏 been 😀 written to 💦✌ be 🙋🎮 as accessible as possible 🔝 to an ☺👏 audience without 🚫🚫 in-depth 👏 knowledge 📚😍 of statistics or programming. This 👈🔴 is 🔥🚫 primarily so 💯 that 👎👅 you 😍🙂 do not have 👃☣ to ✌💦 take our word 🙌 for 👻🎁 its accuracy. 👌 By 😈 reading 📖📲 the analysis, you should 👫😑 be able 💪💪 to 💦👉 understand 😷🤔 at 👌😠 least 👬🚫 on 🔥 a basic level why the ♂✅ statistical corrections we 👥❣ made 🤗😇 account 💳 for 💼 all ✊ the 😤 relevant biases. Additionally, ➕✏ as noted in 🙌 Section 3: 😧 Objectivity, 🤖🤖 we 👦💰 aimed not 🚫 to 😋💦 calculate the ➕💦 precise probability of 💦 Dream 😴 experiencing these 😤 events, but an 〰🦅 upper bound 🤐 on 🔛🏿 the 🅿😈 probability. This 🎄👉 makes 🤔 it ☠🤔 much 💘 more ✋ difficult for bias to 💦💦 have any 💦🍵 effect; 📣📣 if 👏 we 😀💰 correct ✅ for 🌍💕 the 🅱 largest amount 📉 of bias in 👏 the 🎁🚗 data that 🥁 there 🏿 could ❌😈 possibly be, 🏻 there is little 🐩👧 risk our 💩👶 analysis will 👊 be skewed due 👅👅 to 🍆📧 our 💰 bias causing us 🤵 to 💵💦 underestimate how 😱🗣 much 🙀💘 we ought to 💦 correct. ✅ We 👵👨 believe 💭😱 that, to ⏸ the extent any 📨 bias exists, these measures should be more 😏😏 than ☄ sufficient to ✌😤 account 💳💳 for it. 💱🤔 Additionally, ➕ note 📋 that we 👉🤠 are not 🚫 the only 😤 people 😣👥 capable of 🎆 analyzing these 🚑 events—if any 🌐 unbiased third 🤔🤔 party 👌 points 😘 out 😧📤 a 👌🥇 flaw in our 😻📸 statistical analysis or 🍆 notes 😚 a 👌 glitch that could potentially cause 🎗 these 💦🥜 events, they would, 💀 of course, be 😝💰 taken 🚀 seriously. 😒 What if 👏 Dream’s 💭 luck was balanced out ⚔ by 👷😗 getting bad 😤😩 luck 🍀🍀 off 😍 stream? This 👁 argument 🙅 is 💦😞 sort of 👍💯 similar to 👍 the 💰👈 gambler’s fallacy. Essentially, what 👏😱 happened to ♂ Dream 💭 at 🍆🍆 any time outside 😏🔥 of the 🎅👏 streams in 📥 question is 👏😩 entirely 💯 irrelevant to the 👀🍆 calculations 📊 we 👧 are 🅱♀ doing. 😧🏃 Getting bad 😞😩 luck at one 😥♿ point 🈯 in time 🕘 does not 🙅 make good 👌 luck 😟🏼 at a 👌🅰 different 💰💰 point ⬆⬆ in 🖕👏 time more 💯 likely. We 👥👓 do 😫 care about 👏 how 🤷👉 many 🏼💯 times 🤔🤔 he has streamed, since those 👉 are 👄 additional opportunities for 😏 Dream 💤💤 to 💦😣 have 🈶🌈 been noticed getting ➡ extremely 💛 lucky, 🍀 and 💰 if 🚀 he 😡👦 had gotten 🅱💦 similarly lucky 🍀🍀 during one 💯😫 of 💦💦 those 😘😖 streams an 🍑 investigation still 👉 would 💞 have 💴 occurred. However, what luck Dream actually 🤔 got 🏻😩 in 🛌 any 💦🍵 other 👳 instance 💯👉 is 🙀 irrelevant to this 😷 analysis, as it has 🤔👉 absolutely 💯🙅 no bearing on how 💯 likely the luck was 🔙 in this ⁉ instance. 🙄 7 💯💯 7 💯 The Binomial Distribution Note: 📝📋 If the 🌊🚀 reader is equipped with 👯👏 a ✨ basic understanding of 👏💦 statistical analysis and 💰🚄 the 🏼 binomial distribution, they ♂😈 may 🐝 skip to Section 8: 🅱✊ Addressing Bias. Note that 🤔😐 the ⛓🏻 explanations 📝📝 present 🎀🎀 here 💦💪 are sufficient for the 🔭 probability calculations 📊📉 performed throughout the rest 🚔🍑 of the 👽 paper, but 😮☝ are 💯 not 👎 exhaustive. Supplemental reading 📖📃 is 🙄❌ provided via 💰💰 footnotes where relevant. 7.1 💯 The Intuition Informally, if the outcome of a 👌 particular event can be described as "it either happens or 💁 it doesn’t", then 🤔 it 😐 can 🔫😡 be modeled with 😉 the 👏 binomial distributionc . For 👧🍆 example, imagine 💦👑 we 😊 wanted 👩 to 👉💦 compute the odds of 🏻 flipping a 💰 fair 😤👒 coind 10 times and 😘💛 having 😋 it 😏🥇 land ⬇ on heads exactly 😉😉 6 🤔 of those 🐥🐥 times. 🍆 Since a 😎👌 coin either lands on 🚟⬇ heads or it 🥊 doesn’t, we 👍👶 can 💦🗑 use 👏⚒ the 😫 formula for 🐻 the 🚟🎁 binomial distributione to determine the 🏆 chance 😨 of this 👈🍆 occurring. Since we 👨😂 flip the 💰👇 coin 10 🔟 times, 😆⏰ we say 🙊 푛 = 10, 🤑😰 and 💰 since 👨 we 💣😺 want exactly 6 💪 of 👀 those 👞🤔 flips to be 💎👼 heads, 🐵 푘 = 6. ❓ The ☝ chance of 🔴💦 a 👌💰 (fair) 👒 coin landing on 🔥 heads is 50%, so 푝 = 0.5. ➖ If we plug these 👳👈 values 👪💰 into 🤓 the binomial distribution formula, we 👨 get 🔟 P (6; 👧❗ 0.5, 😊 10) 😂 = 10 6 👆 0.5 6 💪🏠 (1 − 0.5) 😊💦 10−6 ≈ 0.205 👌 (1) ➗ To 💦 interpret this ❗ value, if ♂ we flip a ✋ coin 10 🔳💯 times, 🍆🕒 we 👦🌊 can expect 🤗🤗 to 💦 get exactly 😉 6 🤔 heads 🙊💤 about 👂☝ 20.5% 🆗🎉 of 🔝 the 👆🍆 time. 😵💯 To understand why 😳🤔 this 😂 formula yields the 👏 probability of 🍒💦 a binomial distribution, and 🙅 how 👹 to 👮⏬ generalize it, ✔ we 👧👮 break 🙇 down 👇👩 each 👋👋 term. 7.2 💯 Generalizing the 🌈 Binomial Distribution Generically, the probability of exactly 푘 successes with 👏😋 probability 푝 occurring in 👉 푛 trials (in 👏 our 💩 earlier example, 🔥🔥 푘 = 6 🕕 heads with 🎉🍨 probability 푝 = 0.5 💦😏 occurring in 🔝 푛 = 10 💯 flips) is 🔥 given 👈⤴ by ⏩ P (푘; 푝, 푛) = 푛 푘 푝 푘 (1 − 푝) 푛−푘 (2) ♀ We 😃 can deconstruct this 💯📉 formula term-by-term to understand why 😡 this represents the 📲 probability. Basically, 👎👎 this 👈 formula figures out how 💯 many ❔ distinct orderings of 🕯🅱 푘 successes and ➕➕ 푛 − 푘 failures meet 💯 the criteria, and 👏🤔 then 💯 sums the 👨😫 probability of 💯🏻 each orderingf . The 🎁🔝 notation 푛 푘 , read 📖 as 🏿💰 "푛 choose 푘", represents the binomial coefficientg , which is 🍏 the 🍆 number 🔢 of 🔴 ways ➕✔ we ❤🙋 can 🔫😠 observe 푘 successes in 🌤 푛 trials—the number 💦📱 of 🤖 ways, 💫💫 with 💰👏 푛 options for 🎅 trials ⚖⚖ to 💦 be successes, you ☠ could 🤷 "choose" 📥📥 푘 of 🗼👀 them. For example, there are 💨 two ✌ ways to 🐵⏸ observe 푘 = 1 👸 heads 💤 in 푛 = 2 coin flips. The head could occur on 🔛🔛 the first 🔢 flip, or 😩💁 it could 🤔 occur 👻👻 on the second 🕐🕐 flip. Therefore, 👏🎉 2 1 is 👮💦 equivalent to 👌 2. 🙈 With similar 💯 reasoning, 4 💦✌ 2 is equivalent to 6; ❗ there cThe binomial distribution also ➕ requires the assumption that ❗ we 👦👬 are ♀ observing discrete independent 🙅 random variables. Since 👨💦 piglin bartering and ➕ blaze drops 💦⬇ are 🏄 discrete independent random variables (see 👀👀 Section 9: Code Analysis), we ⚡ can 🔫 safely 🚦 make ✋ this 👈🎅 assumption. There ↗ are ⭐🙏 other 🙅✉ considerations about 💦 stopping 🆘🆘 rules which 😡 will 👙👏 be 📖🐝 addressed in 📥 Section 📦 8: 👊⚡ Addressing Bias. dA 👨😗 "fair coin" is 🔥💦 defined as 🍑🕛 one whose 🌄🌄 probability of 💦💦 landing on 🔛👇 heads 🐵 is 🔥 exactly 😉 the 💊🚟 same as 🏿🍑 its probability of 😤 landing on 🔛🔛 tails. We ♂ are 👶♀ also 👨 not considering the 🏽👏 probability that 🚟 the 👀👨 coin lands on 👋🔛 its 👤 side, which is entirely 👐 negligible for ⚠ this introductory-level explanation to 🗣✋ the 🔥👉 binomial distribution. ehttps://en.wikipedia.org/wiki/Binomial_distribution fFor an explanation of 💦👅 why ❓❓ this 👉 works, 💦 see 👀 https://www.youtube.com/watch?v=QE2uR6Z-NcU. 🐕 ghttps://en.wikipedia.org/wiki/Binomial_coefficient are 👶 6 unique ways 💯 to distribute 2 💦 successes (heads) 💤🙈 across 💰 4 trials ⚖ (coin flips). (These 🤤 are 🔢 1&2, 1&3, ♂ 1&4, 👷 2&3, 2&4, 💦❤ and 🤠 3&4.) As the first 👆 term represents the 💲👏 number 📟 of distinct orderings, the 👉 next 📅☃ two 💘 terms represent ✊ the 💦🏻 probability of ⛄💦 any 💦 one ♿ order. To find 🔎 this 🙋👏 probability, we simply take 🐥 the product of 💦 the probabilities of the 👏 events necessary to produce a 💰👌 given 👤⤴ ordering; that 😐 is, 💦 the product 👟👟 of the 👏👏 probability of 🚨👄 observing 푘 successes and 푛 − 푘 failures. Since 푝 is 🚟👮 the 🚧 probability of 💦 a ♂ given 👤 trial being 😑 successful, 📈📈 and there ✔ are 💓 푘 successful trials, 👨 we 👴👵 can account 💳 for the 💞🏾 successful 📈💪 trials 👨 with 🤝 the term 푝 푘 (푝 multiplied by itself 👈👈 푘 times)h ⌚😩 . Similarly, we 👌👦 account for 👷 the failures by 😈👨 raising 🅰🔝 the 🦏 probability of a 👏 failure to the power of 👏 the number 🎦 of 💦🌈 failures. As the 👑 only two 💏 possibilities 💡 in 👏 a 👏➡ given ⤵ trial are 🍑 success ☺🤑 and ☺ failure, and 👏 the 👩 probabilities must 💰🙋 sum 👀 to 1, the 👥♂ probability of a 👌 failure is ☎😜 (1 ♀ − 푝). It 🕘🍆 follows that, since 👨👨 each 👋👋 trial that is 🈁 not a 🍆🍞 success 💰💰 must be 🥜 a 🅰 failure, the 👏 number 🔢📞 of 🐶👨 failures is (푛 − 푘). Thus, the 💰💦 final term is (1 🕴 − 푝) 푛−푘 . Multiplying all 👌🤷 three terms together yields the 👏 probability of 🔴 a 💐 binomial distribution with 😂 a 💰👀 given 👤⤴ 푘, 푝, and 🎅 푛. 7.3 💯⏰ The Cumulative Distribution Function (CDF) It would be helpful 😲🤔 to 💦💦 have 💰👏 a 💰 way to 👏♂ compute the probability of 💀 observing 푘 or more successes. Intuitively, we can expect 🤗🤗 the 🚟😂 probability of observing exactly 푘 successes in 👮 푛 trials 👨👨 to ✌ be smaller than 😽💰 the 😼🤠 probability we 👥🏻 observe 푘 or 😤💰 more successes in the 💌🆘 same 🖕 푛 trials. Referring back 👌 to the 🌀👦 coin-flipping example, 🔥 if we 💏 wanted to 💦💰 compute the probability of ☹🏻 observing 6 or 🚫💰 more 😥 heads within 🎉 10 trials, 👨 then we ♂♂ can 💦💦 simply add 👈 together 😭🏿 the probabilities of 💦😔 observing exactly 😉😉 6 👆🤘 heads, 🐵 exactly 7 ❗⏰ heads, (...), exactly 😉😉 10 💯 heads, 🐵🐵 given by Õ 10 푘=6 ❗❗ 10 🅾 푘 0.5 😏➖ 푘 (1 🤜 − 0.5) 😲❌ 10−푘 😂 ≈ 0.377 ➖ (3) 😩 Indeed, this 👀👈 agrees with 😗 our intuition; it 😤 makes sense that it ❗☠ is more 👏💦 likely to get 6, 👧👆 7, 8, 9, 🈂 or 💀🍆 10 heads 😂 in 10 🅾 flips, than ♀🔪 it 💨 is 🈁 to ➡ get 🉐 exactly 😉😉 6 👏 heads in 📥👇 10 flips. The 👩 chance of receiving 푘 or 🙅 more 👆 successes is 🅱🍆 often 💰💰 referred to 💦💦 as 🛠👦 a ➡👏 푝-value. 💵 More specifically, 푝-values 👪💰 are 🚥 the chance ♂😨 of 🐙😤 observing 푘 or 👱🅱 more successes given the 👏😶 null hypothesis. While ⏳👶 that 👑 nuance is 🏻🗓 irrelevant if you 👈🗣 already 😃 know 🔞 for 🍆🌍 a fact 🏫 the 👨👏 coin is 🔥 fair, 😆✔ it 😩 is important 😍 to 😅💯 keep in 👌👏 mind 😲🤔 in this 😎👇 scenario—our entire goal ⚽😫 is, 😍 essentially, to 💦💦 analyze whether 📊 or not 😥 Dream 💭 is 👮💦 using 🏻🏻 a 💰🙀 biased coin. Armed 💪💪 with a 🏿 basic 🌑🚂 understanding of 💦 the binomial distribution, we 👨❤ will 🅱 now 😱🎅 discuss how 💯 this initial calculation must 😾 be corrected in order 📑 to 💦✌ be applied to 👉💦 Dream’s 💭💭 runs. 💰💰 hFor an 😤🤗 explanation of why this 😰👏 works, 👷 see 👁👁 https://www.youtube.com/watch?v=xSc4oLA9e8o. 😮 9 8 👊👊 Addressing Bias There ✔💦 are ❓ a 🏿 few 😋🔢 assumptions of 🔟 the 🅾 binomial distribution that 😟🔇 are 💰 violated 🍑 in this 👈 sample, some 🈯 of ☹😊 which were 👶 noted in 👏💉 the 😍👏 document Dream 💭💭 published 🤓 on October 27. This 👈 section 📦 accounts for ♿🤙 these 😍😱 violated assumptions, and 👏 proves computations that 😩💰 account 💳 for these 🚑🍆 biases. Note 👋 that 🤔 some 💯 of 🏿😤 these 😤 biases only apply to 😂 pearls, as 😅 blaze rod 🍆🍆 drops 😲💦 were 👶 examined in 😜👌 the same 💩👤 streams as pearls 🍬 due to the 🙆 pearl odds, which are 🏃🔢 independent 🙅 of the 🐐🅱 blaze rod drop ⤵ rate. This 👏 eliminates the 🚮 sampling bias from 💰 the decision to investigate the ✈🍃 pearl odds based 👌🤰 on 👇 the 🔯 fact that they 👩👧 are 💰 particularly lucky. 🍀 8.1 ✊🤔 Accounting for Optional Stopping 🆘🆘 The initial 💰 calculation for 🍆🍆 the 👏🎁 푝-value 💵💵 assumed that barters and rod 🍆 drops 💦 within sequences of 🥗💰 streams are 💩 binomially distributed, which 👏 is 😧💰 not 😅🏼 precisely true 🍆‼ (although 😛😛 likely a 💬👌 very ☣😔 good 👏👀 approximation). For the 🔚 data 💾 to be 👄 binomially distributed, the 🕵 stopping 🆘 rule—the 👨⚖ rule ⚖ by 😈👏 which 🎓👏 you 👨 decide 😱😱 when to 💦 stop 👮👋 collecting data—must 💰 be 👬🐝 independent of 💰 the contents of 🍒💦 the data. 📉📉 For instance, Dream may 📅 be more ⬆ likely to 💦 stop ✋✋ streaming for 🍆👨 the 🏻 day 🕑 after 🕑🅰 getting 😧😚 a particularly good 🏽 run, 🏃🏃 which is 🔁💦 more 🙅 likely to happen ♂♂ on ☝ a 📝 run 🏃 with 😍😍 good barters and ✊👏 blaze rods. Indeed, Dream 💭 did 🍆 stop speedrunning 1.16 🌅 RSG after 😡 achieving a 🐀👏 new 💌 personal best time. This ❓👆 will 🎤🙏 result in the 👏 data 💰💰 being at 👈 least slightly biased towards ⛪ showing better luck 🍀 for 🍅 Dream, 💭💭 and 👅 thus the data 📊📉 is not 🚫♂ perfectly binomial. To 👂⚔ account 💳💳 for 🅱 the 🗣😈 stopping 🆘🆘 rule, we will 🐼 correct ✅ for 👧🍀 the 🅱🌫 worst 👹👹 possible 🔝 (most 💯 biased) stopping 🆘 rule. Imagine 😎 that this investigation was being 🐝👏 conducted by Shifty Sam, a ✝😂 malicious investigator who is trying as hard ⛰ as 🏃 possible 🔝🔝 to 👉 report misleading data 💰💰 that 😐 will 👏 frame Dream. 💤💭 Since 👨 a 🙏👌 lower 😎 푝-value 💵💵 is ℹ more ❌ damning, Shifty Sam computes the cumulative 푝-value 👇💵 after 👀 every 💯 barter or 🕍 after every 🔪ⓜ blaze kill, 💀 and ☑💯 stops ❌ collecting datai once 🍆 he deems the 🛣 푝-value 💵 "low enough" 💦 to make 🖕 the 😹👏 strongest case 😎💯 against 🔫😤 Dream. 💭 This 😞😣 is 💦🔥 the 🍆 worst 👹👏 possible 🔝 stopping 🆘🆘 rule, since Shifty Sam will stop 🏿🤔 collecting data 📊 once 🔂 the 🌊👊 푝-value 💵 is arbitrarily 🤔 low 👇 enough (as 🎣 deemed by 🎨 him to be most 👉 convincing). It should be 🏳 abundantly clear 🔎 that this stopping 🆘 rule 👑 is far worse 🤢 than 🤢 whatever 🏿 stopping 🆘 rule ⚖⚖ Dream actually 👉 followed 😣 during 🚣 his 🥐🤔 runs. It may 🌌 not be 🎮🙋 immediately 👏 obvious how we 👩🌊 can ❓😬 calculate a 푝-value 💵 under ⬇ this 👮🏄 stopping 🆘🆘 rule. We 💰 cannot look directly ➡ at the number of 💦💦 success in 📅👌 the 👧➡ data, as that is 🔥 always 👌👉 going ▶🍆 to 😂 be 🐝 exceptional to 👏 this 👈 degree. What ❓👉 we can 💦 consider, ☺ however, 🤔🖐 is how 🤔🅱 quickly Shifty Sam reached 🕶 his 푝-value 💵 cutoff. Intuitively, we might 🅱♀ expect 🤗 Shifty Sam to spend a 💰 long 📏 time ‼⏱ waiting ⌚ for 🍆🔜 the data 💰 to reach his 😤😤 푝-value 💵💵 cutoff. To 🏼 put it another 🤒 way, it would 💯 certainly be 🐝 surprising, regardless of 🏿☹ how ⁉ shifty Sam is, 💰 to hear ✋ that 😐🗳 Dream got 🍸🎁 30 ✈ successful barters in 👇 a 🔫 row 💦 as soon 🔜 as 😱🍑 Shifty Sam started ▶ looking at 👸 the ⚕ data. 💰 Knowing 💭🤔 that 😩 Shifty Sam only 🕦🤠 decided to 😂 show 👨 you 💯 this data 📉 because it 💯💯 supported ✔ his 🅱‼ argument ♂🙅 would 😎 not really 😆🌈 make 🙋😬 that 🅱 any less ➖ surprising (concerns about 🍾 sampling bias aside—those 😤😤 will 👫🎬 be 💦🐝 addressed later). 🕑 Since 👨 the data reaching 👉 a 🍑 푝-value 💵 this 🐸 extreme so soon is 💯 somewhat surprising even 😂 if 💦👏 we know 😭💭 the 😆 data comes from 😂😲 Shifty Sam, we will 😜⚽ look at 🍆 the 🙌 probability that 😩🔕 Shifty Sam stops ✋ collecting data 💰 at 👨🍆 least as 🕘🅰 soon 🔜 as 👦 Dream 💭 stopped. ⚠ In 👏♀ other 🏭 words, if 🤔🅱 푛 is 💦✅ the 👏 number 😧☎ of trials 👨👨 in 👈 Dream’s 💭 data, our corrected 푝-value 💵👇 will 👏👏 be 😤 the probability that 👉 a series of 🔍 trials 👨 will, 🏼 at 🗽 any point 👇👉 on 🔛👇 or prior 🔙🔙 to 👅 the 푛th 🎃🥖 trial, have a binomial CDF 푝-value at 👨❤ least 🚫 as 🍑🏿 small 👌⏬ as the 👏 one 😫 for 👏🎁 Dream’s 💭 data. 🤓📊 iSince Shifty Sam here 😇 is 🅱 supposed 👏 to represent ✊ whatever 👆 caused Dream 💭💭 to choose 📥📥 to 🎀🏻 stop running 🏃🚫 1.16 RSG, suppose Shifty Sam is, say, Dream’s 💭💭 manager, and 💰➕ can 🔫 tell 💬😲 Dream 💭💭 when 🍑 to stop 🛑✋ or 🅱 continue 🔕 streaming. ⛵ 10 😂 Although 😛 that value 💵💵 could 🤔 be computed through 👉⏬ brute force, 🌕🏼 that approach would involve evaluating the 👏 probability and 🎉 푝-values 👪 for well 🤷 over 😈😏 2 😳 305 different 🈯👱 sequences—which is 🗓☝ obviously 🎳🙄 computationally intractable. As 🍑 such, 😆 we 🏃 used 🚟 a 😍😗 method that 🍆 allowed for 🔄 dealing 👦 with multiple sequences at 😂 once. The 🅱👉 exact algorithm is 💦 somewhat involved, so 💯 a 🌍😰 description 👿👿 has been included in 🚪 Appendix B 🅰😇 for interested 👅👅 readers. 8.2 ⚡ Sampling Bias in 🛌🖕 Stream 💦 Selection As 🙇 mentioned previously, we 🔫 chose to analyze Dream’s runs 💰 from 👉 the point ⬆👉 that 😐👏 he ♂ returned to streaming 😭😭 rather than 😻 all of 💰 his ‼💦 runs 💰 due to ♀➡ a 🅰🏻 belief that, 🤔 if 🤔👏 he 👉 cheated, 💏 it was 👨 likely from the 😄😈 point 👉⬆ of his return to 💦💦 streaming rather than 🅰 from 👊 his 💦👿 first ☝🥇 run. 😱 Although 😛 we 😱 cannot 🚫 be 🍆 entirely 👐👐 certain, 🤔🤔 it 👌 is 🚟😨 also ➕😨 likely that 😷😯 MinecrAvenger decided 🤔🤔 to 💦😱 investigate Dream’s 💭💭 streams due 👅👅 to 👏💰 noticing that ☠ they were 😉 unusually lucky. 🍀🍀 This, 🚮 of 🤤💦 course, 😂 means 🙄😏 that the streams investigated are not actually 😥 a 💰 true random 🔀 sample. Even ☎ if 👏 MinecrAvenger somehow 😆😆 chose streams to 👍 investigate 👏👏 at 🍆 complete 🚫 random, 🎲 we 💏🏼 are choosing to 💦🛏 investigate these 🈷🈷 streams due to the fact 📕 that 🍆 they 🙋 are lucky. 🤞 Thus, we 👨 cannot 😡🚫 treat this 🚙⬇ as a ➡🖼 true random 🔀 sample. To 🏻🏻 account for 👅 the 👏🌷 maximum possible amount 📉 of 🐣 sampling bias, imagine 🤔🤔 that Shifty Sam inspected every 👏 speedrun stream 💦 done by 🔥😈 Dream 💭 and 🌬 reported 🔫 whatever 💯 sequence of 💦 consecutive streams was the 😷 most ⬆ suspicious.j This would 🌨😵 produce the 👘😫 strongest possible 🔝 bias—or at 🍆🤠 least 😱😴 a 🐝☝ bias much 😂 stronger 💪 than there ✔💍 actually 😤 is—from 😂😍 the ✨ choice 😜 of 🐣 these particular Dream streams. Recall the example 🔥 of 🌹💰 investigating the 🤡🐐 20 🔳 back-to-back 😰 heads within 100 coin flips from 🅱 earlier. Much 😂💘 like 👋 you 👍👦 could calculate the 👏 probability of 🚑 20 consecutive heads 🐵🙉 occurring at any point in the 🅱 100 💯💯 flips, we 👨👦 can 🗑🔫 calculate the probability that ✔💄 Dream 💭💭 experienced 🤳 bartering luck 🍀 this 👏 unlikely in 👏🙌 any 💦 series 💓 of 💦🤓 consecutive streams. This 👈📣 would 👉 account 💳💳 for 💦 the 👏 bias from ➡ Shifty Sam, and 💛😮 thus 🏻 more 🙅➕ than account for ❓ the actual bias under ⬇ consideration. To calculate the 🔑🏥 chance 😱 that ⚪⚠ at 👌💯 least 👏😈 one sequence of 🙌😊 streams is 💯🌈 this lucky, 🍀 we 👧 first calculate the ✈🌜 chance ♂🚫 that 😟☘ no sequence is. 👏💦 Assuming independence, we 👦 can 🗑💪 do 👺 this by 😆 taking the 🅱 chance 🚫 that 🤢 a 🎮🍉 given ⤵👤 sequence isn’t sufficiently lucky (1 − 푝) to 💦🙌 the power 🏼 of the number 😯❤ of sequences, 푚. If 👏 an event 👐👐 occurs more 😩 than 😻 zero 👰👰 times, 🕐😆 then ❓👱 it 😏 must 🙋👏 have 😩👌 occurred at ❤🤣 least 👌 once, so 🆘 we can then 🙄➡ subtract (1 🥈 − 푝) 푚 from 🙃 one 🙏☝ to get 😛🍑 the 🎺🌊 chance that it occurs at least 🚫🚫 once, 🅱 giving 👸👸 1 − (1 − 푝) 푚. The number 🔢❤ of consecutive sequences consisting of at 😔🍆 least two 💏 streams from 😮💰 a 🅰 set of 😰 푛 streams is 👌👏 푛 2 , as you 👆 choose 📥📥 two 🎄 different 💰 streams to 👌👁 be 🤔 the 🔑👨 first ☝ and 🏽 last. Adding in 😏 the 푛 sequences consisting of 💦🐲 only one 🏻 stream, 💦💦 which ♀ were 👶 not 🚫🚫 included because 🏽🤔 the 👶🆘 first and last ⬅ stream 💦💦 are the 👨👏 same 🖕😂 stream, 💦 you get 😷 푛 2 + 푛 which 🏼👌 is equal to 푛(푛+1) ❄👸 2 . We 👦😍 can 🔫❗ now ❔ get an upper bound 푝푛 on ☝ the 🌎 푝-value across 👉👏 푛 streams, using 🤳 the ⬆ 푝-value 💵👇 derived 🔜🔜 from 👉💥 our 🌍 sample. 푝푛 ≤ 1 👊 − (1 💸 − 푝) 푛(푛+1) 👸 2 🏻 (4) 💦💦 At this point, 🈯📌 let 💂 us 👫 go ♂🏾 back and 👏🅱 analyze an 😚 earlier assumption we 👶🤔 made: 🏠💰 that 😤 the ♂🕍 푝-values 🅱💰 between sequences of 💦 streams are 😟 independent 🙅 of one another. 👯 This 👁👈 assumption is 😠 false—however, 👳👳 it 😢⌨ is 🙀👏 not ♂ false ❌ in a ☝ way ☝ that could cause 푝푛 to be ✅ greater than this ⬆ upper bound. 🤐 Consider the 😶👌 exact 👌 way 👟↕ in ⬇ which the sequences of 📆🏿 streams are 🔢 dependent on 🔛 one 😈🏼 another. Since 👨 they 🏼😕 all contain streams from the 👏🐺 same 🖕😯 set 📚➿ (those from Dream), some 🤔👨 of 💦 the 💰 data 💰📉 in 😂 each 👏👋 sequence will 👏💯 be 🅰👨 identical to 💱 that 💝 in other 👪 sequences. This 😋 lowers the chance 🚫 that 🔍 Shifty Sam jWe can 💦 safely 🚦🚦 assume ♀♀ the streams reported 🔫 would 🤕 be 🐝 consecutive—it would be extremely 💯😂 obvious that the 🔪😱 streams were 🙈👶 cherry-picked 🍞🍒 if ☔ Shifty Sam reported 👮🔫 the 👻 luck 🤞🍀 in, say, Dream’s 💤 first, 🥇 seventh, and 💦👌 tenth streams. Non-consecutive streams could 🔮 be 👏 reported 🔫👮 credibly in 👏👏 unusual circumstances, ❌ but that 👇🍆 possibility is essentially negligible. could find misleading data, as 🍑🍑 he 👨👨 has 🛒 less data 💰 to look 👀 through 🗺 for unlikely events. In technical terms, we 🔨♀ can 🔫 say 😵😩 the 🛩 푝-values 💰👪 of 💦🌾 the sequences of 😏 streams are positively dependent upon one ☝💯 another—they 🚪🏊 are 🚟🅱 positively correlated with each 👏👏 other. 💰 For 🏔 this 🔥 bound 🤐🤐 to fail, 🤧🤧 the 🏼⤴ sequences would ✅ need 👌 to be ❄🏻 negatively dependent. 8.3 Sampling Bias in Runner Selection In addition to 💦💰 these particular streams of 💦💦 Dream’s 💭 being analyzed due 👅👅 to their high 📓 proportion of 😂⛄ pearl barters, Dream was 🏻👏 initially analyzed out 🌌🉐 of 💦💦 all runners due 👅 to 😅 his experiencing unusually good 👼 luck. 😄😟 Much 😩 like ♂😄 we 📌 calculated 🚜🚜 the ⤴ chance of 💦👮 observing data as 👦 unlikely as 🤔💯 the data in ♂👏 question in any 🔥👏 sequence of streams, we ❣ will 👏💰 analyze the 👽 probability of observing data this 👁 unlikely from any 🌐 runner in ⬇ the Minecraft speedrunning community, using 📤 the same 🏆🤷 formula for the 🚑🐆 chance 🙅 of 💦 something 😅😳 occurring at 💯🍆 least ❗ once 💯💯 in a series 💓💓 of 💦👨 trials that 🏾🍜 we 👧👦 used 📅😏 earlier. This 😂 results 🔢 in 🛌 the following correction, where 푝푛 is 👅😝 the 푝-value corrected for 🍆 a community 👩👩 with 🙌👏 푛 runners, and 💰 푝 is 🌈 the 🌈 푝-value 💵💵 for Dream 💤 in 📥👸 particular: 푝푛 ≤ 1 🤜👀 − (1 ⛈ − 푝) 푛 (5) 🍆 Note 🎵 that, 😐 as 🅱🍑 we are 🔄⭐ discussing the 👩🕜 푝-value 💵💵 for 💦 data this ⬆ unlikely occurring to a runner within their 🍆⬅ entire 👏👏 speedrunning career, the 👻 size of their career is not 😖 relevant. Although 😛😛 a 🎁 runner may 🗓 be more ➕ likely to ⏸ experience 😋😋 six exceptionally lucky 🍀🍀 streams if 😂🤔 they 👥 stream 💦 more often, we 👬 already account 💳💳 for 👏 the amount they 👴🤷 stream when calculating 푝—in 🅱 other 👪 words, 🐎 if 🅱 someone 🕵👬 streams more 💯😩 than 💉⬆ Dream, 💭💭 they 😱 would 💀👌 need a 💻↘ luckier sequence of 👀💦 streams to have 👍♂ an 👅🏻 equally low 푝. 8.4 👊 P-hacking 👮 Perhaps 🤔🤔 Shifty Sam examined multiple types 🅱🅱 of 💦 random 🔀 events and 🍒💰 only ☝👃 picked the most 💯👥 significant ones. 💯 For 🤔🍆 instance, 👉 there could have 😤🅰 been 💴💫 analyses of flint drops 💦 or 🎡 iron golem drops, 💦💦 and ➕ only ☝💋 pearls and 😫 rods were reported ♂♂ due to those 👉 being the 👦👏 most 👉 significant—indeed, some 🍌 other 💰 barter items, as 🍑🍑 well 🤷🤕 as eye 👁😉 of 💦🏻 ender breaking rates, 💰 actually 🚟🚟 were 🙈 recorded. To ✌🅱 correct ✅✅ for 😣 this, 👈🏋 we take 🖐 the 👏🙀 probability of finding 🕵 each result at 🍴👉 least 🤸👌 once among 💰 an 🤔 upper bound ℎ on 👍🔛 the 👀💲 different ↔🈯 types 🅱🅱 of events that 🚟😐 could have 💯 been 👦🥜 analyzed. Unfortunately, 😭 the correction used 🚟🙄 for 👨🍆 selection across ➡ individuals and 👴👏 streams will 😳 not 🚫 work here. That 👋 correction requires either independent or 😤💰 positively dependent probabilities; however, 🤔💰 there 👌💾 are negatively dependent probabilities involved here. For 🔙🍆 instance, 🤔 the 👨 more pearl barters you 👆 receive, the 👏 less 😔 opportunities there are 😱💢 to 👮 receive 👉 an obsidian barter: your 👉👈 numbers of 🤑👏 pearl and obsidian barters are 🙏😊 negatively correlated. We can 💦 still correct ✅✅ for this, 👆 but 😠 it 💦 will 📌🤤 require 📜📜 a much 😩 looser upper bound than 😻 the 🤣 ones 💯💯 we 🏃 have 🈶😤 used 🙄🚟 previously. Remember that 🍆 the ⚰ probability of 👏 any one of a number 📱 of mutually exclusive events occurring is ✅👌 the sum of 💦 their 🍮 probabilities—for example, the chance of 💦💦 rolling 😋😋 either a 👌👀 two 🎄✌ or a five on a 🎉 six-sided die 🚦😪 is 🔥😩 1 6 ❗ + 1 😎 6 ❓ = 2 6 👆💪 . However, 🖐 this is 😂 not the 😦🚗 case 👅🤔 for 🎅🎁 non-mutually 🏆 exclusive events. Consider ☺🤔 the 💰 chance 🙅😨 of rolling either 😌 a 💰 number ❤📱 less than three 💁 or 🚻🙂 an 👹💶 even 🕚☎ number. The 🖥😃 chance 🙅😱 of 🔥 rolling a 🀄 number 🔢 less 📉 than 😽 three (1 👸 or ➕👉 2) is 🔥 2 6 ❗👧 and 👈 the chance 🚫 of rolling 💊 an even number (2, 🕔💦 4, or 🔮 6) ❓ is 😩 3 😗 6 💪❓ . Adding these 🍆 together 👫😄 would 👌 produce 5 ♥🏼 6 💪 . But this 👉 counts rolling 💊 a two ✌💏 twice, 👀✌ producing a number 🎦😧 higher than ⬆ the 👏🌌 true 💯 probability of 💦🔥 4 🏽💦 6 🕕❗ . This 👈 double-counting problem 🏻 is the 👧 reason ♀♀ why ⁉ adding together 👫👬 fails ⛔ for probabilities that 💖 are ❓💥 not 😖 mutually exclusive, so 🆙 it is 🗓 not a problem 🏻 that 😐 our probabilities are 🅱 not mutually exclusive: 12 the 🚟🌜 sum 👁👁 of 👏 the 🌧⚰ probabilities will 👏 still work 💵 as an ✒ upper bound. Thus, we 😱 have ⚠😎 the 🅱 followingk , where 😾🌎 푝ℎ is 😳 the 👊👏 푝-value corrected for ℎ comparisons, and 🌚 푝 is the 😂📱 initial 푝-value: 💵👇 푝ℎ ≤ 푝ℎ (6) We will ⚽ choose 📥 values 👪 for 🍆👅 these formulas and ➕ compute the 😱💦 final 😪🌠 results 🔢🔢 in 😏🏽 Part 💔 IV. However, 🤔 to ensure 💰💰 these computations are ♀ not 🚫 invalid due 👅👅 to 🔎 unusual behavior of 🗜 Minecraft’s random 🎲 number 💦😧 generation, 👪 we 👧 will 👊🅱 first 🏻 analyze Minecraft’s code. kThis is 💥👏 commonly known 💫 as 🙇 the 👏 Bonferroni correction. 13 😏😏 9 Code 😲😤 Analysis When 👌😂 discussing probabilities this 👁 low, ⬇👇 concerns about edge-case ⚔🗡 behavior in Minecraft’s random number generator ⁉ (RNG) are relevant. We 👨 have 👏♂ been working 👷 under ⬇ the ⛓ assumption that the results of 👪 piglin bartering and blaze drops 💦😲 are independent random 🔀🔀 variables, as 🍑 one would 😎 naively expect 🤗 if 👩😂 Minecraft’s ☄🍑 RNG were 👌👶 truly ⚡ random. 🎲 This would 👪💀 mean 🤔 that 🏻🙅 the 👧👏 variables cannot 👊 affect one another; that is, 💦🈶 past piglin barters and 👏 blaze drops tell 📟🗣 you 😕🏿 precisely nothing about future 🎆 ones. 💚 However, 💰🖐 it 💦 may 🤷📅 seem possible 🔝 that, 😐👉 in some 🐔 edge ⚔ cases, 💼 piglin barters or blaze drops fail 🤧🤧 independence in ways ✔🤔 which ✌👏 increase 💳💳 the 🗣🍫 probability of 🐣💦 observing Dream’s 😴💤 data. 💰💰 Here, 😶 we will 💍⚽ analyze how likely that 😝🍑 is 🈁 by 😈 inspecting Minecraft’s ⛏☄ code. 😲😲 Before 🍑😂 beginning 🆕😍 the 🕜❤ analysis, it 😽😉 is 🔥🍆 worth 💰💵 noting that 🙇🚟 if Minecraft’s RNG were 🍑😫 to ✌ fail ☠ in such a 😬 way ↕😇 that 😩💦 piglin barters and blaze drops could 🤔🤔 not 😠 be said 🗣 to 💦🔢 be 🏻👄 approximately ⭕ independent, 🙅 it 💧😩 would 😏🍆 still 🛑😻 be 📖 astonishingly unlikely for them 👬🎊 to fail in ⤵😜 exactly 😉😉 the ❤👏 way required to produce the observed data. 💾💰 The failure(s) would need to 😥👉 (1) occur repeatedly over the 🚗🍑 course 🏎 of 💦 six separate play sessions for 😏👨 Dream, (2) only 👨 occur 👻 to 👊➡ Dream 💭 out of all 😮😩 runners, (3) 😗 affect both bartering and 👏💦 blaze drops, 😲 and 👏 (4) 🕓 specifically 🔵🔵 bias the results 🔢 towards ⛪ piglins bartering ender pearls 🍬🍬 and 🍆🚄 blazes dropping blaze rods, rather ☑🙇 than 🔪🔺 towards some 💵 other barter item or blazes not ⛔♂ dropping rods. Although 😛 this 👈👌 may 🗓🗓 still be more 🤔💦 likely than the 👏👏 data 💰 occurring without a 🃏👦 flaw in 👉👏 Minecraft’s 🍑 RNG, even before analyzing the 🤥👢 code 😤 it 😩🙅 appears a 🅰 priori extremely unlikely. 9.1 Confirming the 👨💦 Probabilities Though 💥💭 the 🕍👏 probabilities we 😺 have ✊ been 🤤 using 🏻 thus 🕵 far 🌌 for 🕓 piglin and 🤔💦 blaze drop ⚰👇 rates 💯😂 in 👏 Minecraft ⛏🚨 1.16.1 🕴 are 🔢 publicly available 💢💢 information, 📚 it is important 😍 to 💰👌 identify exactly 😉😉 where 🌎🤷 these probabilities come 💦💧 from. 👉 The piglin bartering proportions are 🅱 determined by the piglin_bartering.json file 📂 found 🤔🔎 in the 👉🌎 1.16.1 🕛🏫 jar filel . As expected, exactly once 🏳 each 👏👏 barter, the 👨 game 🔥 selects an 😍👹 item from 💥💰 the 😜🙀 following weighted table: 🎲 Item Weight Book 💯 5 😂 Iron Boots 👞 8 ✊ Potion 10 Splash Potion 🍾 10 🔟 Iron Nugget 10 Nether Quartz 20 🔳 Glowstone Dust 20 🎊 Magma Cream 🍨🍨 20 Item Weight Ender Pearl 20 🔳 String 20 📊📊 Fire 🔥 Charge 40 Gravel 40 Leather 🐄🐄 40 Nether Brick 40 Obsidian 40 Crying 😣 Obsidian 40 Soul 😱 Sand 🏝 40 Table 1: 🗿 The 🌌 simplified contents of 🚨 piglin_bartering.json. Here 🍒 an item of 😓 weight 푛 is 👎🅱 푛 times more ✋🍗 likely than 🅰🔪 an 💶👏 item of 🛢💦 weight one. 😉 Additional information 📚 regarding enchantments, stack 📚📚 sizes, and 👏 potion 🍾🍾 effects not 😡❌ shown. 🚫 Since 👨👨 the 🏽👌 weights sum to 💦🌱 423, and 🎅 ender pearls have 🎁😑 a weight of 💦 20, 🔳🎉 the 👏 probability of 💦🔴 an ender pearl barter is ♻🅰 indeed 20 🔳🎊 423 as 🏿 expected (in 1.16.1, 👸 the 👉👏 version 👧 Dream 💭💭 used). 🎶 lTo read these 🔫🚟 files 📁📁 on 🏽 Windows, simply ⤵😡 rename 1.16.1.jar 🌸 to 💰💦 1.16.1.zip 🤜 and ⏱ navigate to data\minecraft\loot_tables. 💰💰 14 👦 Blaze drops are specified by a ❔🍒 file 📁 called blaze.json, an 👴 excerpt of which is 🙌 included below: 😫 1 " function ": " minecraft : set_count ", 2 💕 " count 💯🙌 ": { 3 👏🎆 "min": 🕑 0.0 , 4 "max": 1.0 , 5 " type ⌨✍ ": " minecraft 🍑🚨 : uniform " 6 🤔 } One 👆🤓 can 🔫 see 🙉 that, 👨👨 when the 👏🍤 player’s 🎮💰 weapon 🗡‼ does not ♀❌ have 👏 a looting enchantment, blazes select ❇ between 🏻 dropping either 😤😬 0 or 1 ❌⏰ rods using a uniform distribution. Thus, a rod 🍆🍆 drop 👇⚰ occurs with 🍨👏 probability 0.5 as expected. 🍆 9.2 💦 Setting 🌃 RNG Seeds Failures of 🌈👏 one of Minecraft’s ⛏☄ RNGs to 💦💱 behave randomly are 🏄🈶 not ❌ unheard of—the 💦🤤 most ☺💯 famous 😎😎 examples of 🍳💦 these 🚑❌ are ♂🙏 the RNG manipulation exploits found in 🔙➡ versions prior 🔙🔙 to 👌👌 1.13. 🕴👂 These ☀📀 all work on 👋 the ‼ same principle: 👴🏾 some part of Minecraft’s code resets an RNG being used 📅🆒 by 😈😈 other parts of the 🏠👇 code, causing predictable behavior. 😦

Fallacies And Foundations of Science

• Gamblers Fallacy
• Misuse of Stats
• Data Dredging (Important for us)

Probability Tools/Videos

• Probability Trees
• Law of Large Numbers or: Why Number of Rolls Matters
• Discrete Overview (MIT Video)
• Conditional Probability (MIT Video)
• Example of Conditional Probability (MIT Video)
• Multiplication Rule (MIT Video)
• How we calculate the probability of a sequence of events (MIT Videos)

Gambling Help

• National Helpline

Edit: don’t take it too seriously friends! Have fun. If you’re going to post up some analysis, just make sure it passes a sniff test. Statistics are a powerful tool, and easy to misunderstand and misrepresent. :)
EditEdit: I see how the title could come across as harsh or rude -- definitely not my intention! I think the assertion made there is ridiculous, but I don't think anyone is stupid for coming to that conclusion and I definitely am not trying to judge that redditor.

Some incursions are inbound, soon, so I need to post this for you guys. This post is for new guys, if you already successfully make money with options, don't read this post and fuck up your vibe.
I want to keep this short and sweet.
​
Most of the people in here are what I call "vibe" traders, I include myself in the group. I have seen some truly incredibly gifted technical traders. These people are geniuses, the read the markets like Neo reads code in the matrix. I will never be one of those people. In my trading experience, I've met 2 of them.
So! How do we do this right, if we are not technical geniuses.
Here are the rules.

1. Understand that most options portfolios will go to 0. Even if you are quite smart. This allows us to have the correct attitude from the start. We are not entitled to a win, we are betting against a very smart house that has more experience than us. When we lose, it is because that is the nature of things. It is not anyone else's fault.
2. Have a vibe. This is the beginning. Don't jump into someone else's vibe. Use the data around you to confirm or refute your vibe.
3. Verify your vibe. Look around for data that supports you. This is where tools like NOPE, forums, this sub, news, past performance etc are valuable. Jumping on a play because a tool or someone else told you to is a recipe for disaster.
4. Challenge your vibe. Look for data that refutes you, using the same tools as above.
5. Know how to make an entrance. If you jump in when everybody else is jumping in, you're going to lose. Namsilat's deeply offensive rant about basic TA is a good place to start. Don' buy at the top of an occilation. https://www.reddit.com/thecorporationwiki/comments/j8vj2d/the_lost_writings_of_nam_silat_chapter_one/
6. Act decisively. Enter your position the very moment you think you should. Don't wait to see if you're right.
7. Stop out of losses. Set a percentage that you're willing to lose, leave the second you hit it. Don't wait to see if your loss is "confirmed"
8. Take profits. Set a percentage gain ahead of time to leave your position.
9. If you hit it big, get out of the game. If you make 100 grand, get out. You're an outlier and the nature of the gamblers fallacy is about to eat you. Buy some shares if you want to keep playing. We'll all miss you, but I'll feel much better knowing you got away with some life changing money.

Recent example from my own past. I hate Intel. In my mind they are everything that is wrong with corporate culture. Take a successful engineering company, silence the engineers, try to make it a successful "financial engineering" company. I won't ever go long on it because it violates my vibe. So I ignored what spurious data was saying and bought puts.
​
Buuuuut. I don't hate Netflix quite as much. I actually like them. Now, my brain told me that the "Cuties controversy", pandemic reopening, and massive competition by everyone in the world was going to be bearish for them. But I ignored myself, ignored my own rules, and entered a bullish Netflix play for earnings. Instead of using the tools available to confirm my own initial beliefs, I used them to open a position COUNTER to my beliefs. I deserved to lose money and I did.
​
Be nice to people trying to help us, stay humble, take responsibility for your own losses and don't fight the market.
Oh, last thing. Right now, holding overnight is gambling. It's a fine thing to gamble, but be aware that that's exactly what you're doing. Flipping a coin.
​
Thanks for being here,
-fO

Too busy celebrating Caillou's cancellation to play (nah, just kidding, studies are the culprit) but I can squeeze some time out now to continue the game a bit! Just finished the Gabomba Statue dungeon, so I'll take a look around Kibombo and see if anything's changed before revisiting places where I can use Piers' Frost (for now, I remember the Madra Catacombs and the Kibombo Mountains).
-Seems like everything's peaceful again around here, but to be honest I'm not optimistic at all about Akafubu's leadership being anything but pathetic. Who knows, maybe if I keep coming back to Kibombo I'll be able to see character growth. Is there a teleporting Psynergy in this game?
-Speaking of Akafubu, his dad's already singing his praises, but Akafubu's only beating himself up for leaving the room so quickly - not quite sure if he's doing it out of greed or something else, tbh.
-All the warriors were farm boys? Sounds pretty horrible for the town's economy and sustainability, but what do I know? It's literally like the military draft, I guess. Also, the weapon and armor shopkeepers talk about deadly poison and 'maaaaagic' being infused in the warriors' spears, but all the war paint in the world isn't gonna help you against a fireball, bud.
-Speaking of weapons and armor... Jenna (ironically) nabs the Frost Wand for herself, and that's it for weapons. As for Armor, Piers gets an Iron Helm and some Gauntlets. After some item shuffling (and seriously, what are these Game Tickets even used for?) I clear up enough inventory space so that Jenna can get some Gauntlets too. Item lady's got some daaark thoughts.
-...I should've guessed that monsters were eaten in this world, tbh. Thinking about it, how do people even separate normal animals from monsters here? What's the difference between, say, a wolf animal and the something-Wolf enemy from all the way back around the Kandorean Temple? Or the Emu monster and the actual emu? Food for thought... and for the stomach.
-Ooooh, some guy's talking about there being treasure west of the river... but he also thinks that nobody in Kibombo's ever found it. Well, mister, I guess they just didn't have superpowers. A woman in another house says that they're blocking the way across the river, and that you'd need a ship, which is convenient for us - we've got the Black Gem and we've got Piers, so we'll probably be able to get his ship soon enough unless anything goes wrong. West from Southern Gondowan (pretty sure that's where we are right now?) is the Eastern See, which is sorta confusing - supposedly, there's two continents in the middle of that sea. Guessing we'll go there soon enough. Found a Lucky Medal in a jar in the same house, which is nice. It seems like the villagers are starting to be more positive towards Akafubu, so we'll see.
-That 'being nice to people training' thing sounds hilarious. I want to watch that.
-Huh, I thought there'd be something on the furthest south point of the cliff you need to use Frost to get to. Well, going up a screen.
-...Wait, this is the exact same place where we recruited Piers - we just jumped on a roof instead of using the ice pillar. Whoops.
-Well, the Gabomba Statue's still open, so... yeah, I might as well try the dungeon the Great Gabomba opened last time.
-Gabomba Catacombs... hmm. The grass here reminds me of the one in Pokemon where wild Pokemon could attack you. There's also a crater-thing in the ground that won't budge even after I try all the field Psynergy I've got. I also run into a couple of Red Demons... which reminds me that someone commented to try and kill them with a Jupiter Djinn to get an item, so I'll try that. I think this is the first time I'm actually using the Defend command in this game, wow.
-Well... holy cow, that was a lot of XP. But no item dropped, so... unless anything interesting happens, I won't update anything until I get an item drop from the demons.
-Having to constantly re-set the one Jupiter attacking Djinn you have is a bit of a pain. Really wish there was a way to make fleeing more reliable too.
-jeeeeesus fucking christ, these salamanders' psynergy spells are stupid strong. or maybe Sheba's just stupid fragile.
-Jeeeeeeeesus, the Vital Moon left Piers at O N E HP.
-Pretty sure I've killed more than 16 Red Demons with Blitz already...
-Yep, this has officially entered gambler's fallacy territory.
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-F i n a l L Y
-Staff of Anubis, huh? All right - holy crap, that took longer than I expected. Oh my god. I spent waaay too much time running away and failing and then running away and succeeding and then healing up with Felix, but everyone's level 20 now. The new staff is a power boost... but not exactly a big one, haha. Would I do it if I knew? Probably not. But here we are, and at least I've finally got it. Don't think I can do anything right now in the Gabomba Catacombs, so let's get out.
-Oh my god, the innkeeper's wife is offering BBQ Monster Steak. That sounds sick - in both senses of the word. The warriors seem like good guys now that, you know, there's no raids and storming-cities and sneaking-into-sacred-rituals-shenanigans going on. A short rest, and it's time to leave Kibombo.
-Into the Kibombo mountains! I remember seeing a Djinni in here - let's see if Frost can help us.
-Oh my fucking god
-The Sarcophagus unleash literally summons a literal sarcophagus that crushes the pixie flat like a pancake. That's hilarious, 100% worth it, I now regret nothing.
-Oh, a Growth plant! It's annoying to have to switch to another class to use the Psynergy, tbh, but them's the breaks. I save before interacting with the Djinni, and thank goodness, because it leaps into a fight. After a few spells traded, Sheba whacks the Djinni into submission. I guess Waft must've been impressed with that, because it goes straight into Sheba right after that. With that, she class changes from a Magician to a Mage - nice!
-Felix levels up to 21, and I decide to go to Naribwe. Let's see what to do next with the fortune teller... and besides, I'd also like to see what else the people living there have to say. I run into a Wyvern Chick on the way - and man, I used to be so scared of them, wow. I still am.
-Seems like some soldiers came through Naribwe asking for Piers... that doesn't bode well. Don't think the Kibombo knew his name, so I'm not quite sure who they might be. Otherwise, a guy's got updated dialogue, but still thinks that Kibombo's a dangerous place at the moment - I suppose the signal flare and the messengers haven't gotten to Naribwe just yet.
-...Holy shit, the innkeeper's husband dropped a bomb: it's Isaac! Isaac's trying to get to Piers! Not expected, holy crap - urgggggh, how did Felix miss them? And why do Isaac and company want to talk with Piers, anyways? Does it have something to do with the Lemurian boat Babi took from the country and gave to Isaac...? Huh. Food for thought.
-Seems like the girl that mentioned Magma Rock to Felix also mentioned it to another traveller, who headed in that direction. Might that be Isaac? But she didn't mention a group, so... not sure. Another kid mentions a man trying to get to Kibombo - I think that one might've been Piers, actually. So I guess that we're actually ahead of Isaac's group right now... huh. Usually the playable characters are the ones doing the pursuing, so this is a nice change of pace.
-Seems like the fortune teller's been predicting some pretty apocalyptic things. "The power of nature would rekindle the advance of civilization," huh? Interesting. The lady gets philosophical afterwards, asking whether a peaceful or advanced world would be better.
-...Wait, what? The old man in the backyard we broke into to nab the Unicorn Ring mentions that a 'big brute' came by looking for, of all people, Menardi? How many freaking people have been to this village? Pretty sure Saturos died with Menardi when they fell into the hole on top of Venus Lighthouse, but... hmm.
-Awww, the Kibombo are trying so hard, but I guess it's been a while since they've had to communicate like that, huh? Doesn't seem like anyone in Naribwe really understands what they're doing - though the fortune teller did tell them that the Kibombo's warlike ways are over, so let's see what ends up happening. The priest mentions a freezing cold region called Tundaria, all the way down in the south. I'll keep it in mind too.
-Time to have a reading or two. Setting down Sheba's Staff of Anubis leads the fortune teller to say that "beyond the weapon I've set down lies a terrible foe," and that my next foe awaits in the misty sea. I have to gather the "pieces of the weapon" (whatever that means), or drown in defeat. Next, I set down Piers' Chainmail: to the north, beneath the sacred icon, the one I seek awaits. Pretty sure that means in the Gabomba Catacombs, but I don't think I can progress there just yet (I should know, I spent way too long running around that floor). Setting down an Elixir tells me a similar thing to the staff: that I must forge a weapon to defeat a mortal enemy, and to seek out the pieces. I feel a fetch quest incoming. I try to set down the bag with the Jupiter Star, but I "can't remove it", so that's that, I guess.
-Well, that was a very informative diversion. Glad I stopped by here, it's given quite a lot of things to consider and watch out for. Time to move on! I think next was... Gondowan Cliffs?
-Right, Gondowan Cliffs. Pretty twisty area. There was a puddle somewhere...
-Found it. I use Frost and use the pillar to go from the right to the left. There's only one drop, and to the right of the cliff below is a Healing Fungus that I picked up. Kinda like the Laughing Fungus I've got right now, actually - I think I got that one here too. There's another puddle near the other entrance, but I backtrack after following the path because I'm pretty sure it's little more than a shortcut.
-Back to Madra! For... the fourth time, I think. Let's check out what the folks around here are saying, and then venture into the catacombs.
-I see they've fully covered up the catacombs entrance behind the graveyard. I should probably do the catacombs before I leave Madra so I don't potentially miss anything good. The guards are in high spirits, and inform us that the mayor's returned! Let's see how the guy's holding up.
-Quite a few people are talking about the boat in Alhafra: they mostly think that it should be ignored or destroyed, which seems kinda short-sighted, but hey. An old man in front of the vendor stands is instead thinking about Lemurland Lulu Lemuria.
-...There's a woman with a portrait asking for Menardi. Let's just say 'no' for now, haha... um. She can detect the Mind Read Psynergy, so she's an adept, but she doesn't accept us telling her that Menardi's dead, so... hmm. Well, at least Felix has told her the truth; hopefully none of these decisions come back to bite me on the ass later.
--Oh, so the Shin guy - the item vendor's boyfriend - is the guy that was being an ass to Piers and got some Frost Psynergy demonstrations for his trouble, huh. Not sure what to say other than 'don't be rude', haha.
-Well, the other vendors are still thinking about the first raid. Seems like the armor vendor doesn't think that there was an alliance because they attacked from different sides - and to be honest, given that Briggs' ship got wrecked on the right of Madra, it would've been hard to coordinate with the Kibombo even without their warlike attitude. Coincidence sounds kind of fishy, though... was there somebody pulling the strings?
-Some girl in the cave house is talking about a business scheme that's gonna use Isaac's power. Whatever that is, it doesn't sound very good. The guy in the house is thinking the same thing; to be honest, I do like that there's all sorts of varied reactions. This greedy fuck wants to parade Isaac around like a circus animal to get some money, and I could sadly see that happening in real life, given what people have been shown as circus freaks in the past. But more importantly - Isaac's group was here, huh? Was it before or after going to Naribwe?
-One of the sailors, Isaac... The guy even mentions the Lemurian ship he was given. Doesn't seem like those guys in that house see him as very trustworthy. A kid below the house I was just in actually clarifies - seems like they're looking for Lemuria. Did they mention that in the first game? I don't remember - but they're probably going there on Babi's orders, either way.
-A woman in the church is fearful of the future, pointing out the ocean warming up and the fish dying. Seems like nature's going to shit - kinda fits in with what the fortune teller supposedly said earlier, according to the lady in Naribwe. The people in the house next to the church are instead talking about the Kibombo; seems like they think they'll attack again. Hm. Can't blame them.
-The lady in the house with the Djinni's talking about mushrooms from the Gondowan Cliffs now, which certainly catches my attention: I smell a sidequest. I've got the option to hand him something... so I'll save first. Uh, the Laughing Fungus has the description 'a rare and suspicious mushroom', so I'm guessing that's the 'bad ending' to this sidequest? I'll save, hand it over, and probably reload.
-Ah, so it's not quite right. Well, that's better than him eating it and keeling over dead with his face in a puddle of his own vomit. It was returned to Jenna's inventory, so... huh. Well, let's hand in the Healing Fungus then.
-Seems like this is it. Glad to be of help to the old man: everyone deserves some comfort in their lives. They're going upstairs... and yep, they're fetching the Mars Djinni. Seems like they had it as a pet - which brings up a question. Non-Adepts can see and interact with Djinni, but can they actually use their unleashed powers or gain something from them? Regardless of the answer, Jenna gets Char for herself, and everyone's happy: I get a Djinni, and this old couple can have some tasty mushrooms for dinner.
-Going down to the prison before going to the mayor's house, and it seems like the mayor got fed up with the greedy bureocracy in Alhafra, which... 100% fairs, yeah. He wants to get people back there himself to fix the boat - I'm guessing we'll help him in that endeavor sometime. More confirmation that Isaac's party showed off their Psynergy: seems like one of them even took the time to explain what it was. Whoever it was, I'm guessing it wasn't Garet, haha.
-Ahaha, some people in the prison are trying to freeze the puddle. Glad to see they're taking it better than the guy in the cave house behind the stalls, anyways: this seems benign (and tbh, I'd definitely try too). Seems like Shin's trying to better himself - I'm glad. Time to go to the mayor's house, I guess - I'll pop into the inn to mind-read the people there after coming back from the Madra Catacombs.
-Oh, everyone's coming out for this one, nice. Everything worked out fine in the end, woop! Sheba calls attention to the girl outside (which doesn't surprise me, considering her mutterings about Menardi and her standoffish behaviour) and is swiftly ignored. Seems like Piers wants to go to Lemuria, just like Isaac - oh god, they're gonna be chasing after us trying to get information, aren't they? Do they even know that Felix and company are with Piers now?
-Awwwww, Piers is saying that he thinks we're all good people now! That's great, haha. Is he counting Kraden? I really hope that these guys get some dialogue, because I think there's a hell of a lot of potential dynamics that could be explored. I'd love some interactions between Sheba and Jenna, for example. Seems like Piers has decided to lend us his ship so that we can get to Lemuria... nice. The group promises to visit, and the conversation's over. I just have to highlight the old lady's thoughts, though: she thinks Piers would've been eaten by monsters if he'd been alone. Lots of faith.
-The mayor stops us when we leave to give us the gift he'd promised: a Cyclone Chip. Interesting name. I'd totally forgotten about the gift, tbh: you're all right, mayor :)
-...uh oh, "Felix...?" and random evil music? Seems like this girl knows us.
-Goddamn, Sheba - blunt as shit. "Isaac killed them. They're probably at the bottom of the sea by now." OOOOF
-Seems like the party wants to protect Isaac, which is very sweet and noble, but Menardi's sister is smart enough to figure out who he is anyways, which sucks. Kraden suddenly pipes up with "How do you know that Felix didn't kill your sister?", which is proooooobably not the best thing to say unless you want Felix to die. Is that it, Kraden? Do you want Felix to die? Menardi's sister calls him out on his bullshit (but seriously, what the hell, Kraden?) and says that even at double power, Felix'd still get stomped, which... fairs. She also says that "the lives he hold dear hangs in the balance," which... hmm.
-All right, so Menardi's sister is gonna try and seek out Isaac while Felix's group lights up Jupiter Lighthouse. She mentions "we" - is the big brute mentioned earlier that was also looking for Menardi with her? She refers to herself as Karst. Cool name.
-"Wow. And I thought Saturos and Menardi had issues." I LOVE YOU SHEBA
-The group's actually intelligent (shocker in an RPG, I know) and discusses the ramifications of their conversation. They catch on to the fact that Karst said 'we', and discuss the possibility of warning Isaac before dismissing it; and I can see why, since their goals are literally opposing each others'. Besides, Isaac's already seeking us out, and Karst is seeking him out... if she saw us fraternizing with the 'enemy', I don't think she'd exactly be merciful.
-Heh, Sheba calls Isaac and Jenna an item. How long ago was the Sol Sanctum incident, anyways? "...Stupid Sheba..." Jenna mutters. I love these two - they've already gotten so comfortable around each other that they can joke about silly stuff like this. It's great. Kraden also brings up that if a confrontation came, they'd squash us... but I dunno, iirc I finished the game with everyone at level 30 or 29 and Felix is already level 21. A bit more time... and we'd at least be able to fell Ivan, hahaha. Time to venture into the Madra Catacombs before they close for good!
-We progress a bit (the enemies are so puny now, and I'm still not over bees being here) until we reach a door: "Look upon me with eyes of truth", the tablet next to it reads. Using Sheba's Reveal Psynergy, the tablet turns into a button - pressing it opens the gates, and we enter into the ruins. I see a chest on the left and a chest to the right, but can't reach them quite yet. A bit of running around, however, and we get to the chest: it holds an Apple! The jars next to it have nothing, though.
-You know, I never noticed before, but the monsters have higher or lower-pitched death cries: for example, the Drone Bee's is higher-pitched than the Mini-Goblin's. I see an open door, but let's try to get as much business on the outside done as we can. Some Lash action allows us to use the Frost pillar to get above the open door - but there's an open door there too.. I backtrack and check if I missed anything, but going to the left only leads to the Tremor Bit chest I opened a while ago.
-Ground floor entrance: quite a few staircases and doors. The right staircase leads to a room with nothing but an empty box and a destroyed bed. Jenna levels up to 21 after a fight with a couple of Trolls. The left staircase is blocked, huh. The left room has a lot of jars and barrels, but none of them have anything - I do see a chest behind some rocks, though. The right room has a staircase that leads down to a room with a locked door and another staircase that eventually leads to the chest from earlier: it has a Lucky Medal, which Jenna pockets with gusto. Finally, the central hallway has a table cracked in half by a large stone and a seat (a throne?) worn down by the passage of time. Creepy stuff, honestly - what the hell happened here?
-Frost and Lash combined allow us to get to the top level. Going down allows Felix to nab a Mist Potion from the chest that we saw in the start. Restoring 300 HP to everyone is no joke... this could definitely be useful. Entering the first floor's entrance, we can go through the left door now. There's a bedroom inside, along with a bookshelf with a chest. Move doesn't work, but Tremor does - the chest drops below. It's not in the left room, but it is in the room down the central hallway. A Ruin Key... I think this is for the door on the right room's staircase that goes down.
-The Ruin Key works, and there's (what I think is) the final secret of the catacombs - a summon tablet! Moloch's power is ours. Time to test the summon out and then retreat out of here.
-...It's kinda ugly-cute, haha. And with that, we're out of the Madra Catacombs! Time to go to the inn and see what's going on there.
-The innkeeper reassures our party that the fish he uses are fine - his thoughts confirm that, luckily. A lady that works in the inn mentions that someone was asking for a weapon - a trident, she clarifies - that 'he' could use to slay a sea monster. Interesting. Pretty sure the fortune teller told Felix that he'd have to find a special weapon to fight a sea enemy with, too. The merchant up in there (and how long are those two staying, anyways? Like - woah) theorizes that an undersea volcano's responsible for the rising temperatures and the tidal wave. We're probably gonna explore it somehow, watch, haha. Seems like that's pretty much it for the dialogue in Madra, though, so I pay for a night and then prepare to leave.
-Oh, shoot, the Cyclone Chip, I nearly forgot about it. Seems like it gives another Field Psynergy (we're running out of inventory spaaaaaaace). I give it to Jenna: she's got quite a bit of PP, so she should have something other than Lash.
-We leave Madra, but someone calls out after us. It's, uh... one man and two purple aliens. Seems like they know Isaac.
-Oh - OH! Vault; that's where we recruited Ivan, right? And we fought bandits that were trying to steal something from Hammet (Hammett?) and his wife (Lady Lanaya or Layana, don't remember). They want to fight - and holy shit the battle themeeeeeeeeeeeeeeeeeeeeeeeeee I'm dying
-It's so good
-Why are the bandits still purple, though. Like, is that their actual skin colour
-Wow, what a throwback. The bandits are down in a couple of turns, and flee. Kraden says something like "Will you stop trying to get your revenge on Isaac?" after they've gone off-screen, which is... something. But they drop Golden Boots, and holy shit do those make your Agility go up the wazoo. After some consideration, I give them to Piers: the Agility boost would be a bit superfluous, and I do like the idea of Felix being a slow and steady healer. Time to get to Piers' boat! ...Where was it again?
-Oh my god, it's a Wild Wolf. I haven't seen them in foreverrrrr!
-Right, found it - East Indra Shore! There's a puddle, which I freeze and use to get to the cliff: there's a Cookie inside the box, nice! Now it's time to get to the boat itself - when we do so, Piers says that we have to get to the energy chamber below. There's nothing in the barrels and nothing in the crow's nest, so might as well follow. The interior's pretty cozy, huh. A Barrel's got an Elixir, but none of the others have anything, so time to proceed.
-...An Aqua Jelly. There's monsters in here? Ooof. The monster goes down in a few hits, and... dissolves into a puddle of water. Interesting. I check down the stairs to make sure I'm not missing anything, but the way is blocked by some crates, so we can't go anywhere else. Piers levels up to 21 after we defeat the second Aqua Jelly and celebrates by solidifying its remains into ice so that we can get a Potion. Going down.
-Holy crap, that's a lot of Aqua Jellies. There's a lot in the adjacent room, too - what in the world happened here? Some more shenanigans ensue, and we manage to get through to the other side and push the crate down in the hallway for a shortcut. Gotta say, though, this has a pretty horrendous layout for an actual ship that people would use.
-More Jellies and some pipes. You know, all this Frost usage is making me wonder where's the equivalent Fire power. Venus has Move and Growth, Jupiter has Whirlwind and Mercury has Frost, but Mars has... nothing. Some fireballs to light tinder up, or an explosion to blow up big rocks? Oooh, or summoning lava. While I'm at it, I clear out some inventory space by giving Felix a Cookie and Piers an Apple. A box has an Antidote - exciting stuff. No other barrels and boxes that I check have anything though, so I freeze the last Aqua Jelly and move on.
-I pass a few rooms and go into the first Aqua Jelly-filled room we saw, the one with the chest. The barrel's got an Oil Drop (Briggs!!!!!).
-...I didn't save before this
-I didn't know there was a boss. Welp.
-All right. Aqua Hydra, bring it on. I start off by using some Djinn to bolster my attack and defense, while reducing the Aqua Hydra's defense as well. The Raging Flood it uses in retaliation nearly one-shot Sheba, but what's new? I really wish I had some multi-target healing Psynergy right now, though. Ugh.
-Well, Sheba and Jenna are dead thanks to Raging Flood spam, and the only revival Djinni I had was Jenna, so it looks like it's The Boys' Time. The hydra thankfully wastes quite a few turns using Triple Chomp against Piers and dealing single-digit damage, and after a few very tense rounds, one final Ramses summon finishes the enemy off. Phewwwwwww.
-...Well, that chest is now underwater. But at least we can now continue!
-We go down into a really weird chamber that doesn't really vibe with the rest of the ship: some Douse usage opens the door. Piers sets the Black Orb... and everything starts shining - and we're sailing! Piers tells Felix to take the tiller, which Google informs me is "a horizontal bar fitted to the head of a boat's rudder post and used for steering". Jenna wants to go to Lalivero, and Tolbi, and Vale - which I definitely wouldn't be opposed to. Gods, having complete and utter freedom must feel very nice for Sheba, Jenna and especially Felix after so long being trapped with Saturos and Menardi. Piers agrees, and Felix too, but Kraden decides to be the rational one a party-pooper. And with that... holy shit, I'm controlling the ship. Or I guess you could say... holy ship!
-I make a quick stop in the beach near Alhafra to revive Sheba and Jenna (you just have to press A next to the beach, I think, which is pretty convenient). A small fee of 820 coins later, and everyone's ready to go - but it's pretty late here, and I think I've gotten quite a bit done today. It's been a lot of fun, and I can't wait to explore the sea and see (heh) what stuff lies in wait for us.
Everyone's Level 21 except Sheba - she's Level 20. We've got 3 Venus, 4 Mars, 4 Jupiter and 5 Mercury Djinn, and Felix is a Knight, Jenna is a Hex (interesting name), Sheba is a Mage and Piers is a Commander. That Aqua Hydra seriously caught me off guard - thank goodness I just about managed to win. Apart from that, though, this session's been more about plot development and exploration than about combat, which isn't necessarily a bad thing. And now that we have a ship... well, I guess we have to find the parts to make the weapon we need, according to the fortune teller, so hopefully I'm able to find some next time I play.

Yesterday I read this post made by d3on and decided to share some thoughts in the subject explaining a bit more in-depth how some systems are made for swaying you into spending money. I will be linking some other gachas which have some stuff that are not currently in genshin, so if they do appear at some point, you can be aware of why they're there. I will also try to give some advice in how to avoid them, but take them with a grain of salt, because that's just me.
I'm sure you can find all of this information out there in a more concise and with a better written text, but figured i could maybe inform some people and give something back to the community.
*I'm sorry if I make grammatical mistakes, english is not my native language, and I would highly appreciate criticism*
I'm going to start by introducing the concepts, and then showing some examples in-game of how they could be used to make you spend more money, you can jump to any title that catches your attention if you would like.

System 1 and System 2 - Unit Banners and time limited offers

Think about it for a second. Why dont we have all the banners available all the time? Why do we have to wait to get a chance for our favorite character?
When you look at a friends face, you can almost immediatly recognitize the emotions he's/she's feeling, should it be sadness, happiness, frustration, discust and so on. You decodify alot of information without even noticing it, but at the same time you can't quite tell why - you would most likely take a few moments to figure it out. That's what Daniel Kahneman coined as being "System 1", it's a kind of process your mind makes, that is working as long as you're awake, and is trying to understand and make sense the world around you. It is fast, deliberate, (i guess you could call it intuitive), and is proned to make mistakes.
On the other hand, when I ask you, what's 17 * 24?
You know it's a multiplication problem, and you could figure out that the answers 123 or 12,609 are not likely, but you can't be certain it is not 568. You also know that you can solve it using a pencil and a paper, and maybe even without them. But your mind is not at ease when you engage in that activity. It takes effort, and is troublesome, and in most of the time you try to avoid it if you can. It is the type of mental process that if you're driving, it is not something that you should perform while taking a left turn into dense traffic. It is slow, effortful, and requires concentration, that's System 2.
There's alot of gacha games with limited timed offers, summoner wars being a good example, or event unit banners and the purpose of that is to not give you enough time to engage system 2. You need to act under pressure, otherwise you might have to wait a long time until that unit you want comes / you could lose that awesome offer and what not. If only your system 1 is engaged, that means you will not take into as much consideration exactly HOW MUCH that offer is worth. Is it really that good? Is the offer worth that price? Do you NEED it? Would getting that make you happy? These are questions that could make you change your mind, and as such, you should always be considering them before doing any purchases. *If you're in doubt, wait it out*

Availability Heuristic - the 5*

Look at this image and try to answer: What's their profession?
You could say they are law students, wall street traders, just some random people at a corporate job. But, most likely, the answer that came to your mind has to do with a close related experience. Maybe you had to go to your bank so you could unlock your account and keep spending those bucks trying to get Venti. Maybe you had to deal with some nasty stuff and needed a lawyer recently. We tend to remember things that are available in our recent memory.
Look at the rates: 0.6% chance to get a 5*, but what do we see? You go into various discords and there are "Gacha Hell" sections, where you can see loads of people getting multiple 5* even. Everybody getting lucky, f2p accounts that are better than p2w ones. With all of that, you might start overestimating the ACTUAL probability of getting stuff. That might lead you to "do just one more pull" when for a fact you would not do otherwise if you were being constantly reminded of how bad the rates are.

Some irrationality - Really good deals?

Let's say you want to go in a trip. You see the following 2 deals. A trip to Paris, with hotel and meals paid, at a decent price. Another trip, to Rome, again with hotel and meals paid, at the same price. Depending on what you expect from each place, it can be rather difficult to choose where to go. Both cities have really cool places to go, and you can almost surely have a good time in both of them. But lets say i add another offer. Another trip to Rome, but without the paid meals at a very marginally lower price. Most of the people are going to choose the Rome full package in that situation. Having that one "bad offer", makes the "Standard Rome" offer seem more appealing.
When, in gachas, there is one clearly overpriced/bad offer, it is likely that it's serving that porpouse. It makes the other offers look good by comparison, even when they're not. Trying to observe the offer for what it is should be a good way of getting around of this irrationality, then again, engaging your system 2. d3on has highlighted this very nicely when talking about the Blessing of the Welking Moon.

Loss averse, but risk seeking

Problem 1:
In addition to whatever you own, you have been given $1,000. You are now asked to choose one of these options: 50% chance to win $1,000 OR get $500 for sure
Problem 2:
In addition to whatever you own, you have been given $2,000. You are now asked to choose one of these options: 50% chance to lose $1,000 OR lose $500 for sure
Most people choose the sure thing in problem one. On the other hand, most people choose the chance on problem two. We dont like to lose what we have. If you choose the 50% chance on the first one, you would be "losing 500 bucks". If you took the sure thing on problem 2, the same thing. But how does this apply to gacha?
If you did some summons on venti banner, and counted them, and lets say that you did about 50 and got nothing. If you pay for the 40 summons left that guarantee a 5*, and the banner rolls out, you are going to be "loosing" those 50 summons, even if you did get some good stuff on the way. Maybe you did 50 summons, and you think, "I'll just buy a moderate amount, something like 20-30 on the chance that I get Venti, because I dont want to lose my attempts!". Remember that you could need another 90 rolls to get him for sure, as you are not guaranteed on the first roll. You are risk seeking when the odds are agains't you, and you feel like you're losing something. That 50% chance is there for a reason, and I personally think it's really cheap.
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These are some of the things I had to say about it, I hope that it has been helpful to you if you read all the way here. I guess I need to say this as well but, even though there are all these things designed to make money off of you, I still really enjoy playing Genshin and I think it is a great game.
I wont be saying "the company has to make it's money" because there are a number of different ways of doing it, that does not involve employing psychological mecanisms. They dont exist because the company NEEDS it, they exist because the company want as much as it can get (you can take that as you want).
I could write some more about Sunken Cost Fallacy and Gamblers Fallacy giving some more examples, but i feel like d3on has already done a good job at it, nonetheless, if you guys want I can do some more.
Most of the information I've used is from the book Thinking Fast and Slow by Daniel Kahneman, if you guys are interested. This talk is really good if you guys want a briefing of the book.
One example is from the book Predictably Irration by Dan Ariely.
Thank you so much for your time, I'll be trying to answer anything that might come up.

I chose to post here, because my post got removed after I posted it on unpopularopinion
I mean a low IQ probably would affect your ability to make decisions that are right since it affects our ability to understand and apply our knowledge as well as think critically about things, but a higher than average IQ would not necessarily make you a better leader or decision maker.
I formed this thought after reading through The Intelligence Trap by David Robson. Overall the book talks about why talented, knowledgeable, or intelligent people would fail to make the right decision and often end up being stupid.
Their are many reasons why intelligent people can be stupid. But the main reason is that IQ typically doesn't measure many of the skills, temperaments, and cognitive abilities required for good leadership or decision making, including the ability to avoid cognitive biases in thinking. IQ isn't a holistic measure of intelligence or ability.
Also, intelligent people often fall prey to certain traps. According to book by David Robson, their are three types of mentalities that "trap" intelligent people or educated people.

• We may lack the necessary tacit knowledge and counterfactual thinking that are essential for executing a plan and preempting the consequences of you actions
• We may suffer from dysrationalia, motivated reasoning, and the bias blind spot, which allow us to rationalize and perpetuate our mistakes, without recognizing the flaws in our own thinking. This results in our building "logic-tight compartments" around our beliefs without considering available evidence
• We may place too much confidence in our own judgement, thanks to earned dogmatism, so that we no longer perceive our limitations and overreach our abilities
• Thanks to our expertise, we may employ entrenched automatic behaviors that render us oblivious to the obvious warning signs that disaster i looming, and more susceptible to bias

And one more that I'll add

• We often fall prey to cognitive biases, such as the outcome bias and the gamblers fallacy

Also, it's possible that having a mental illness would have a effect on your decision making (although it does no necessarily affect your leadership capability). Example include when a person in the manic phase of their bipolar acting impulsively or a person with Schizophrenia doing things based on their delusions and hallucinations.

Hello all, I have revisited the infinite monkey theorem and have been giving it much thought as of late. I have come to the conclusion that the infinite monkey theorem is false. Upon sharing this information with many people online they have all simply made like monkeys and threw their feces at my face saying I'm just an ignorant idiot (Apart from a select few). So I have decided to share my thoughts on Reddit in hopes to gather a larger consensus.
Firstly I will start with presenting the theorem; The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare. In fact, the monkey would almost surely type every possible finite text an infinite number of times.
Ignoring the fact that a real monkey would just slam a rock on the typewriter and take a shit on it, the actual point is not the monkey, but the random generation of letters eventually producing verbatim, deliberate works of art. Now to be completely clear, I do accept that it is entirely possible for this to occur. It is entirely possible for a magical immortal monkey to physically type out such works given enough time. However, the issue is the term "Almost surely" as if to imply that it must occur.
To start off we must first understand the concept of infinity. Fundamentally it is impossible to think of a string of words or numbers that goes on for infinity, as it is so large that even our brains would not have enough storage space to memorize and account for every number or letter. This is the same in terms of time. Infinity is not just a large value, it is a boundless value. However there can be different types of infinity. For example, any whole number can theoretically have an infinite value, as well as any combination of whole numbers such as 11111 reoccurring and 122222 reoccurring and 133333 reoccurring, and so on to an infinite set of infinite numbers. This can also apply to decimal numbers such as 0.111111 reoccurring, and so on. Similarly this can apply to letters, such as one A followed by an infinite string of B's, one A followed by an infinite string of C's, and so on.
With this in mind I would like to assert that the monkey, given an infinite amount of time, has the ability to string together any infinite combination of possibilities. For example, the monkey could simply type the letter S for an infinite amount of time or any combination of letters which exclude any deliberate piece of writing. It is also possible for a different monkey to type out a string of letters, as I did mention earlier, containing hamlet.
The reason we must accept that it is possible for an outcome to be an infinite string of S, is due to the gamblers fallacy. The probability of any given coin flip does not change simply because it has been tails 12 times in a row. The theoretical probability that S is typed an infinite amount of times draws closer to 0 with every tap of the key, but never actually reaches 0. This is why in probability, the term "almost surely" is used. Even though the mathematical value of a probable outcome is considered to be 0, it does not mean it is impossible. In saying this, it is equally possible that the monkey types an infinite string of G's or B's or any other combination of valueless letters. They all have a mathematical probability of 0.
So how do people argue that the theorem is still correct? Well they assert that when stringing together any infinite combination of 'finite' strings such as the work of hamlet, followed by the complete transcript of the transformers movies, as well as each version of the bible on hand, etc increases the strings value, approaching to a 1 (but again never reaching it). This is to say that it is argued that it is more impossibly probable for any given string to contain hamlet than it is to not contain hamlet. That is to say, there are more infinite sets of hamlet strings than there are non hamlet strings. And when you put it into perspective and think about it in this fashion, it becomes blatantly false.
Finite works, no matter how many, are finite, and any infinite combination of finite works are also finite. In theory you could also have every finite string of letters followed by an infinite string, but it is less likely for any deliberate string to occur than it is for a random string, if not at least the same in terms of probability. For example, it is more probable for a string of letters to contain "zpt" than something like the word "indubitably", meaning it is more probable for a string to contain gibberish than any legible string of words. Another example is that the English language heavily relies on certain letters, the letter E for example appears 11.1607% of the time in the oxford english dictionary. Where as the letter Q only occurs at a percentage of 0.1962%. This makes the letter E 56 times more common than the letter Q. Another example is by the oxford university press who complied a list of the most common words used which analysed over a billion words. "The" is by far the most commonly used word.
If we are to assume that a monkey is to type out all great works of writing in English, then we can assume that the monkey will be heavily relying on typing the letter E and the word "the". Again, the probability is astronomically low, but not impossible. The argument however, is that it is more likely for random letter generator to generate all finite works of writing, than infinite strings of letters. Well, if we are to do some simple math we discover that 100/26 is equal to 3.84615384615. Meaning that each letter has a 3.84615384615% chance to occur. If we revisit the occurrence of the letter E in the English language, we discover that likelihood of producing the letter E at random is far lower than what is required to produce every word. This is again, not to say that there doesn't exist a realm where the letter E is produced at random 11.1607% of the time, and where the letter Q occurs only 0.1962% of the time etc. But that is is far less likely than simply typing any given letter at random, even in the realm of infinities.
Simply stating that an infinite string of "S" to be irrational and invalidating that string on the basis of probability is absurd when the probability of the letter S being typed is far greater than what is required within our English vocabulary. Furthermore, there are an infinite string of letters which can be taken as gibberish which do not contain an 'irrational number'. There are an infinite amount infinite gibberish where hamlet simply doesn't exist. People will say infinity is to hard for people to grasp and that is why they cannot understand hamlet existing within the infinite, but i would say it is the opposite. They are so fixated on the infinite containing the finite, that they forget what infinite truly means.
If this still doesn't convince my readers so far lets then pick apart the argument and demonstrate why it is fundamentally flawed. The argument asserts that: there are an uncountable infinite set of strings which do not end in such a repetition (such as zptzptzptzpt...) these correspond to the irrational numbers. We can then separate these into two uncountable infinite subsets. Those which contain hamlet, and those which do not. However the 'largest' subset are those which do contain hamlet, as well as any other combination of written work.
Well, the basis of this argument relies on what we call irrational numbers. And as such irrational numbers are less likely to be picked than normal numbers. And somehow, it is normal for any given random string to contain hamlet and irrational for it not to. If we actually look at what an irrational number means, we get this definition: In mathematics, the irrational numbers are all the real numbers which are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers.
So if it to be considered a normal number, and not a rational number, it must be expressed as the ratio of two integers. An integer is a number that can be written without a fractional component (aka a whole number). Well, since we are dealing with infinity, any combination of finite work will eventually come to an end in the span of infinity, and thus must have an infinite string attached to the end. This means that while containing finite works approaches us towards 1, it will actually never reach it, meaning there are no strings which are integers, which is to say there are no rational strings, which is to say there are no 'normal' strings, which is to say the probability of the monkey typing a normal number is not 1.
Taking all this into account it is impossible for a string containing hamlet to exist if there is a string which only contains gibberish such a "zptzptzptzpt". Given there are infinite strings which do not contain hamlet, and that hamlet is not a requirement to exist within the string. The equation we actually are looking at is hamlet+∞. And it is far less likely for this to exist than simply ∞. Regardless, the term "almost surely" does not apply, as the monkey will not almost surely produce hamlet.
Who knew infinity wasn't rational.

Now that keyfarming is gaining some traction in the community again, I'd like to have a discussion on keyfarming strategies the community may or may not have developed.
I have more or less read about the standard keyfarming methods:

1. Always go leftwards in Countess Tower
2. Memorize the 3 paths to Nihlathak and recognize the portraits on the walls.

​
And a little more obscure trick involving cutting corners in Arcane Sanc posted by LivEisJeebus:https://www.reddit.com/pathofdiablo/comments/ff1jdz/my_keyfarming_tricks_what_are_yours/fjufihw?utm_source=share&utm_medium=web2x&context=3
Additional discussion in this thread:
https://www.reddit.com/pathofdiablo/comments/ff1jdz/my_keyfarming_tricks_what_are_yours/
I couldn't really get the corner trick to work 100% of the time, if anyone has any tips to get it to work more consistently I'd love to hear them.
​
Which brings me to two personal hypotheses I would like to get feedback on:
Hypothesis #1:
Assuming the Summoner path is generated completely at random, and that the chance is completely independent to prior games (which makes sense), what are the odds of seeing "streaks" in a sample of 100, or 1000 Summoner runs? Example: Directions 1, 2, 3, 4: What are the odds that in a sample of 1000 runs, you will find streaks of 3, of 4, of 5, etc. in the same direction? Like 1, 3, 3, 3, 3, 4, 4, 2, 1, etc.
​
How often will we see a streak of say, direction #3, three times in a row within that sample?
​
Which brings me to the next question: Is it possible to reduce the chance of getting a 4th way summoner to less than 25% of your runs? Currently, I try not to go the same direction as the previous game's summoner direction. My memory isn't good enough to collect all the samples of the hundreds of runs I have done this ladder. On this strategy I have managed a 6 streak first way summoner, but that could very well be RNG. I don't know if going the same way every time will ever yield a 6 streak first way Summoner. It's possible and more likely to be encountered the larger the sample of course.
Edit#1: This is all wrong. It's a common fallacy called "Gambler's Fallacy". I was partially also thinking of "Probability Matching" which has no statistical basis but is commonly observed in nature. Basically, the strategy of PM given, say, 80% chance to flip heads and 20% tails, is to randomly guess 80% head and 20% tails all the time. Which is wrong statistically speaking. You maximize your gain by guessing "heads" all the time. And you will eventually end up with 80% right given enough trials.
Hypothesis #2:Countess tower will never spawn close to the borders of the Black Marsh. So my general teleporting strategy for finding the tower will be to teleport till I hit a wall, and teleport roughly diagonal or away from it till the wall is "just" out of sight, to check out as much of the minimap as possible. Has anyone ever seen Tower spawn close to / on the edges of Black Marsh? How far do you keep away from the edges?
​
I have recorded a 10 keyrun sample on a 200FCR Meteor sorceress which includes Eldritch and Shenk.
https://youtu.be/O0aM1OeAlu0
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Would love to hear any critique on teleporting, or mouse click in general. Figured since I'm spending so much time keyfarming, every second saved will add up in the long run. It seems to be that I spend a lot of time loading in/out of games (approx. 5seconds each game). Not sure how I can improve my performance there. Playing on an old 2013 Macbook Air.
​
10 keyruns over 26 minutes works out to an average of 2 minutes, 36 seconds per key run. Not the best, but I did 4th way the Summoner many times in this sample, on top of a quick break to Akara. I think Eldritch and Shenk adds about 10-15 seconds each run too.
I believe the best run in that sample was Run #7, 2 minutes 12 seconds. My personal best was 1:32 but it was incredibly lucky. I have seen even better ones on 105 FCR sorceress which is very impressive to me, I have a long way to go in terms of reading the countess maps and more precise teleporting through Tower and Arcane.
​
If you made it this far, thanks for reading! Hope to hear some insight on your personal experiences in keyfarming.

The subject of auto-farming comes up once or twice a week in this Reddit and every time it is greeted by the exact same two people:

• Person 1 makes the argument that they do not have the time to invest as heavily into the game, therefore the less that they are able to look at their phone, the more productive they feel they would be in other aspects of the game such as PvP content. They make the argument that many people quit because of a lack of auto-farming.
• Person 2 makes the argument that auto-farming is detrimental to the health of the game and will result in less engagement, therefore more players quitting. Less time spent actively engaged with the game coupled with a quicker depletion of crystals will result in less overall phone time spent on the game.

Both Person 1 and Person 2 are correct, and have valid points to make. However what we rarely talk about are the plain and simple facts that we don't like to acknowledge when we are playing a game, especially one with the Freemium model that so many mobile games have today.

• Com2Us is a business, first and foremost
• Com2Us will make decisions based on maximizing revenue
• Com2Us, despite poor customer service and the general perception of not caring about their player base, did not earn 2 billion dollars in revenue by sitting around with their thumb stuck up their ass

So let's break down the arguments made by both Person 1 and Person 2. Instead of analyzing these from our feelings based on our personal relationship with Summoners War, we are going to instead analyze these from the perspective of a game developer and a business owner. But before we do this, we need to recognize Summoners War for the game that it is, rather than the game we expect it to be.
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The Freemium Game Model, the Sunk Cost Fallacy, Skinner Box Mechanics, and the Feeling Of Getting What What We're Owed.
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The Freemium Game
A quick Google search tells you what a Freemium game is better than most could summarize. "The freemium model offers users the core product — the game — for free and then optionally charges them for premium content such as in-game currency, extra content, or customizations." We are all aware of this and separate ourselves into different factions based on this accordingly - Free To Play players and Pay To Win players (F2P+P2W respectively.) What's interesting about these two factions is how they can so vastly differentiate themselves and the gymnastics we do to convince ourselves what faction it is that we belong to.
By definition, a F2P player will have spent $0 on this game. A P2W player will have spent at minimum $1 to play this game. How many times have you read someone say "F2P btw.. But I only get the $5 package to keep my account secure". If a player has purchased $5 every 15 days from this game for a period of 5 years, this equals roughly $600 spent on this game. Does that sound F2P to you?
Another very common argument is that you are not a P2W player if you only buy certain packs. The transcendence pack every Christmas, that's it. A 750 crystal pack when you were off sick from work and wanted to spend the day farming. Does this sound like F2P to you?
The point I am trying to make within the Freemium Game Model is that we are all capable of making mental gymnastics work in our favor of convincing ourselves that we view this game differently than how we actually play it. Which is a perfect time to talk about the way that mobile game developers and businesses such as Com2Us manipulate our trains of thought using extremely common tactics used in every business from getting ourselves a Starbucks coffee in the morning, to walking into a casino and dropping $1,000 at the slot machine.
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The Sunk Cost Fallacy
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Individuals commit the sunk cost fallacy when they continue a behavior or endeavor as a result of previously invested resources (time, money or effort)
Does this sound familiar? To put it plainly; the more time and money you put into something, the more you feel it owes you. For this point, I like to liken hitting the replay button in the dungeons akin to pulling the lever of a casino slot machine. You sit down at the machine, pull the lever, watch the lights go by. You either get a payout (rare), you get something meager not worth the value of your pull ($1 back on a $5 pull for example), or you get nothing. This is exactly what we are doing with the dungeons. We are pulling the lever (the replay button) and hoping for a big payout (a legendary rune) and scoffing at the meager payouts we deem invaluable (runes that we need to sell) or get frustrated that we receive nothing (unknown scrolls)
The only difference is that a slot machine can pay you out nothing, however a dungeon run cannot. So unknown scrolls and rune craft stones are put in their place to simulate the mechanics of a slot machine adapted for a mobile game.
Now that we've likened dungeon runs to the slot machine, I'd like you to consider how you may behave sitting behind a slot machine. Do you walk from one slot machine to the other and try your luck at different ones over and over? Likely not. You are more likely to sit at the same slot machine and pull the same lever because you have invested your time, resources and money into this slot machine already. This tricks our brains into thinking that the machine is more likely to pay out to us when in fact your odds would be just as good sitting at any other machine. The longer we sit at this machine without a payout, the more frustrated we get, and the harder it is to move away from it. So you keep trying, over and over.
"The effect becomes a fallacy if it's pushing you to do things that are making you unhappy or worse off. "
Sound familiar?
But hold on, Person 2, because this isn't just proving your point. It is also diminishing the main argument that you pose stating that the health of a game is directly tied to whether or not auto-farming exists on that platform, and cite other games that feature auto-farming to have died. Person 1, you likely have heard Person 2 claim that Com2Us has had other games that feature auto-farming die off despite being exact replicas of Summoners War. And why is that? Well, you can blame the Sunk Cost Fallacy for that as well.
Imagine walking into the casino and you begin to make your way over to the slot machine you have been pulling for the past five years. But wait, there's a BRAND NEW, shiny, sparkly, up to date, modernized slot machine sitting dead center in the middle of the casino. It pulls for you! It has better graphics! It has a drink dispenser! It appears to be better in every single way!
.. Good luck pulling everyone off of the slot machine they have spent the last five years pulling, never having quite obtained the payout they were hoping for. This is the manipulation that game developers like Com2Us have over us and that they are well aware of. "Oh, you want to go try something new? lol you'll be back. You've spent too much time here to leave now."
The Sunk Cost Fallacy works both ways. On one hand, it's unfair that one demographic doesn't feel that they get something they perceive as a QoL update that would enhance their experience. On the other hand, the other demographic can't really use it as too strong a basis for their arguments either based on the fact we've ALL sunk time into Summoners War. Other games didn't necessarily die because they didn't have auto-farming - it's just as likely they died due to not being able to pry enough people off their slot machine of choice.
Summoners War is one of the longest running and most successful mobile games in existence. It hasn't survived through dumb luck. It survived because it knew how our minds worked, it exploited that, and now they've got us in their clutches - just like casinos do to the millions of people on this planet who suffer from gambling addictions.
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Skinner Box Mechanics - Why Our Brains Suck, and the Chemical Reason Com2Us Won't Implement Auto-Farm
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https://www.youtube.com/watch?v=tWtvrPTbQ_c
I won't outline the whole history of Skinner Box Mechanics, but I'll summarize as best as I can

• Researchers found that they could condition animals based on having pigeons peck at a button inside of a box. Each time they pecked the button, they would receive food, or, a reward
• Researchers found that this conditioning, called Operant Conditioning, could work on humans
• Researchers found that simply rewarding someone every time they perform an action, is not the best way to get someone to continually do that action. If you provide a reward to the person a random number of times or every number of minutes, these methods are far more effective at getting someone to repeat an action.
• Researchers know that we are well aware of this, and we keep doing it anyway.

I can't say much more on this topic because it is completely black and white.
We are not getting auto-farming because Com2Us is aware that the more they can get us to press a button and reward us some of the time is far more effective than giving us the ability to farm 10x the amount with 10x less effort. It is not a matter of how we think we might feel. It is a matter of how our brains are wired to feel.
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In summary:
Giving us the ability to auto-farm would be completely counter-productive to Com2Us' business model, their bottom line, and the science behind the Sunk Cost Fallacy as well as Skinner Box Mechanics. Our brains are wired to press the replay button over and over, even when we don't feel like doing it, for the chance at a reward that we don't know will ever appear. Com2Us exploits this, just as casino owners exploit this of their gamblers.
Person 1 and Person 2 both have good points to make about having and not having auto-farming. The unfortunate reality is that it doesn't matter what Person 1 or Person 2 thinks, because neither of these have a basis in the conversation as to why Com2Us operates the way that it does. Com2Us will make their business decisions on what will generate the most revenue, not what will affect one majority of the player base over the other - Com2Us would rather have 1,000 players spending $20,000 a year, than have 100,000 players spending $20 a year.
It is up to the individual to be aware of these mechanics and choose accordingly if you are able to view the game for what it is and if you would like to continue playing. If you are living under the illusion that they might implement full auto-farming one day and keep pressing replay because you hope it will come to fruition, I hope this has helped you understand why it is not coming and can help you decide accordingly if this game is harmful to you, or a fun part of your daily routine.
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***EDIT***
Yes I saw the most recent developer note. Welp.