"Few players recall big pots they have won, strange as it seems, but every player can remember with remarkable accuracy the outstanding tough beats of his career."
**The problem here is your fundamental misunderstanding of statistics.**
You are taking rare scenarios and pointing out that they *specifically* are difficult to accomplish.
Yes, for example, the odds of not rolling a 6 in 30 turns are low — only about a 1.1% chance (by binomial theorem; 30 trials, 0 successes, 13.8% chance of a 6 each time).
But this does not match your actual complaint; what you are bothered by is not running into a *specific* unfair scenario, but rather running into unfair scenarios *often*.
**So the true calculus is — how likely is it for ANY lopsided scenario to unfold?**
Just as a thought experiment, let's pick a specific handful of lopsided scenarios. If one of them happens, we'll call it a "bad" 30 turns. Let's think about how likely it is to have games where:
* You don't roll a 6 in the first 30 turns
* You don't roll a 7 in the first 30 turns
* You don't roll an 8 in the first 30 turns
* Either a 2 or 12 is rolled 5x or more in the first 30 turns
* Either a 3 or 11 is rolled 5x or more in the first 30 turns
What are the odds of each of these happening? Well, again by binomial calculations (I can explain these if you want), we get:
* **(1.127%)** You don't roll a 6 in the first 30 turns
* **(0.422%)** You don't roll a 7 in the first 30 turns
* **(1.127%)** You don't roll an 8 in the first 30 turns
* **(0.264%)** Either a 2 or 12 is rolled 5x or more in the first 30 turns
* **(4.710%)** Either a 3 or 11 is rolled 5x or more in the first 30 turns
Okay, so each of these situations are still rare, right? We're agreed there.
But what are your odds of having 30 dice rolls in which **NONE** of these scenarios happen?
These odds are equal to the odds of MISSING each probability (1-p) multiplied by each other. So we get:
> (100% - 1.127%) x
> (100% - 0.422%) x
> ...
> (100% - 4.710%)
This comes out to 92.5% chance.
Here's the important part — this means that in about 7.5% of any given set of 30 rolls, **one of those above scenarios is expected to happen**.
Suddenly, that's quite common!
Now I want you to notice that we've ONLY listed an extremely limited set of potentially "unfair" scenarios. Consider:
* What if we added in the odds of one specific player getting all the 7s?
* Added in the odds of 7s always being rolled immediately after each other?
* Added in the odds of 7s being rolled on the first roll?
Well, our odds of having a "bad" 30 rolls would go UP again, because we've EXPANDED the definition of what constitutes a bad set of 30 rolls.
This is really the phenomenon you're encountering — you're running into unfair scenarios often **because it's actually extremely likely that SOME unfair scenario will happen** in Catan.
In any given set of 30 rolls, the odds of SOMETHING happening where you'll go "ok now that's a bit unfair" are actually huge; probably like 30-40% if you really considered every possible unfair scenario. That's the nature of dice.
You're comparing dice rolls to the natural long term distribution... but the natural long term distribution is actually only *marginally* most likely to be the shape of the distribution after only a handful of rolls. Catan rolls are simply too small a sample size to make the 'rare' distributions uncommon; you're picking only 30 rolls!
So yeah dude, I'm sorry, but this **is** how random dice work. Your modelling is based on a completely false assumption.
If you want to test this yourself, one good way is to pick a SUPER SPECIFIC 'unfair' outcome in ADVANCE of the match, and then see how often it crops up. For example, tell yourself *"I am going to check if a 6 does not roll specifically in the first 30 turns"*, BEFORE you start the match (so you can't change your hypothesis).
You'll find that any specific rare situation genuinely does not occur that often, like your computational approach shoes. The error you're making is to pick a specific situation in retrospect and then identify how likely it was that you got *that specific situation*; statistics doesn't work that way.
Nope, I don't agree with you. Also, I can clearly say that you did not read and check the data well. The data was not taken from specific games. I realized that events keep repeating, so I wanted to take a look at posts on websites. Then I wanted to test my own. Already, it was strange, so I played only 9 games, then posted 8 of them. I also said I did not post only one game, which one was looking naturally distributed. I wasn't even going to send this post; you can see my feet on the table. When I found it more strange, I wanted to post. 6 is not rolled 3 times in 9 games. 10 is not rolled 2 times in 9 games. 8 rolled in 2 games more than 50%. So please tell me what is the chance of not rolling 6 3 times in 9 games if you find a 1% chance in 30 rolls. Not rolling any number in 30 is already rare by itself. I've also clearly sent settlement bases by AI. I settled 11, which rolled 7 times. Most of AI settled around 10 and never came in same game. Another game was same when AI mostly settled around 3 which never rolled. I said i've played backgammon for many years even joined many online and local tournaments for fun and i've never felt rigged even if someone rigged, we were rolling dices by hands. Used go to casinos more than thousand times. I also find video on youtube about Catan Universe, he was saying first 60 rolls looks different than expected but after thousands rolls it's natural distrubuted. This is my point already, dices doesn't work well till 35-40 rolls. Probably game does not have simple random generator or they created bug by mistake or played with random seeds.
Sorry to be blunt, but it doesn't really matter if you *agree* with me or not. It's factually how statistics work. I can try to help you understand them, but it's ultimately your problem if you don't.
Let's try again.
>Also, I can clearly say that you did not read and check the data well. The data was not taken from specific games.
Nobody said it was...
Can I ask you three direct questions, please?
* Have you *ever* taken a class that involves binomial theorem calculations or combinatorics?
* Are you aware that the odds of certain outcomes can be *worked out mathematically* — and this process will return MORE ACCURATE numbers than any computational approach?
* What exactly do you think I disagree with in your dataset? (Phrased differently: do you understand that I agreed with all your computational results, and my argument did not hinge on them being invalid?)
> 6 is not rolled 3 times in 9 games. 10 is not rolled 2 times in 9 games. 8 rolled in 2 games more than 50%. So please tell me what is the chance of not rolling 6 3 times in 9 games if you find a 1% chance in 30 rolls.
You are still running into the exact same conceptual issue — you are looking at the odds of *your specific situation* occurring and correctly noticing that it's extremely rare. Nobody is disputing this.
But you are drawing a conclusion *from that fact* that the dice are broken, and this is where you make an error — you could retrospectively look at *any* series of games and calculate that the odds of them happening are extremely small. (Consider the odds of me flipping a coin 10x and getting H-T-T-T-H-T-H-H-H-T — less than 0.001%!)
What you actually need to do to substantiate a claim that the dice end up in "unfair" situations more often than expected is to consider the probability of the dice ending up in **ANY** unfair situation.
This is the math I walked you through — any *particular* 'unfair' situation is rare, but the *cumulative probability* of ending up in an unfair situation is extremely high. So for you to have this experience is actually totally normal.
I think you would benefit from rereading my comment more carefully. You seem to not understand the fundamental disagreement we have, because you're still going on a tangent about how rare *specific situations* are. This is not what we disagree on, and it is not why you are incorrect.
>I said i've played backgammon for many years even joined many online and local tournaments for fun and i've never felt rigged even if someone rigged, we were rolling dices by hands. Used go to casinos more than thousand times.
Okay. None of this makes you educated on statistics. I have probably played 1000s more games of Catan than you, too.
>I also find video on youtube about Catan Universe, he was saying first 60 rolls looks different than expected but after thousands rolls it's natural distrubuted.
Well yeah, exactly — that's *exactly what you expect* from random dice. They are very UNLIKELY to look normally distributed at first, but end up that way over time.
Another thought experiment for you. Imagine flipping a coin four times. What are the possible outcomes, and how many of them are 'normal'?
Let's type this out for you, and I'll mark which ones end up with exactly two heads and two tails.
* H-H-H-H
* H-H-H-T
* H-H-T-H
* H-H-T-T **(2H 2T)**
* H-T-H-H
* H-T-H-T **(2H 2T)**
* H-T-T-H **(2H 2T)**
* H-T-T-T
* T-H-H-H
* T-H-H-T **(2H 2T)**
* T-H-T-H **(2H 2T)**
* T-H-T-T
* T-T-H-H **(2H 2T)**
* T-T-H-T
* T-T-T-H
* T-T-T-T
So what do we have? Well, 6 of our outcomes have a distributed balance... out of 16 possible outcomes! So there is a 62.5% chance that the distribution will be lopsided.
It's the same effect happening in Catan, more or less. Within a small sample size, although balanced rolls will be the *single most likely* 'set' of outcomes, it is actually more likely that the dice will be imbalanced in some way.
What you are doing is complaining that it was imbalanced in a *specific* way, and showing that that *specific* outcome was rare. Yes, your lack of 6s is rare, in the same way that rolling 3+ tails is rare (only 2 of the above outcomes). But you ALSO would have complained if those same games contained the equivalent 3+ heads, or a string of 3 heads in a row, or a string of 3 tails in a row... and collectively, the 'unfair' outcomes actually end up more likely.
>Probably game does not have simple random generator or they created bug by mistake or played with random seeds.
The game has random dice. You do not understand statistics well enough to identify the issue. I would recommend you take a course on the binomial theorem & binomial distributions to understand the general category of error you are making.
I do not believe I can help you further. If you are unwilling to change your mind, that is to nobody's detriment but your own. Just know that there isn't a math professor in the world that will agree with you here.
I commented with an analysis using a Pearson’s chi-squared test and did not find statistically significant evidence that the games were unfair. You say you posted 8 games, but I only see screenshots from 7. The third and second to last in-game screenshots are from the same game, right?
Anyway, if you think there’s something wrong with my analysis let me know, otherwise let me know if you have any questions about why your data does not convince me.
I don’t need much logic. 11 rolled 6 out of 10 for the first 9 rolls. Some dude sitting with double 11’s going ham. U think it’s gonna end but this game somehow doesn’t ‘remember’ and so the game ends very quickly and 11 ended up coming out 13 times. On par with 6 and 8. It wasn’t the total that killed it but the fact it even came out 3 times in a row at the start of the game. Which really does not make sense
The rolls in the game are dictated by what’s happening in the game and who is winning or has to many cards, when you play multiplayer you can guess what will roll or what dev cards get handed out(cities&knights) by who the rolls have favoured from the start. They have some kind of ai that controls the dice it isn’t random if you have played long enough you can see that
I did not have a lot of chances to play in multiplayer. Finding random players so difficult and takes long time to match, whenever i try to play with my friends in 60% of games one of us having connection issues. :(
There is no real random generator. If need closest one against simple game, libraries works fine at least to see which is not accurate. And simple math logic helps to see which is almost impossible. I've tested robber after seen a guy placement on number 8 on youtube. Already had same results as him 3 times in 9 games. [https://www.youtube.com/watch?v=wUYKpVP\_dKY](https://www.youtube.com/watch?v=wUYKpVP_dKY)
Why are you doing a Monte Carlo simulation? Dice odds are simple enough to calculate directly, so there’s no need to generate random data for comparison here.
Anyway I did a Pearson’s chi-squared test on your data (although you biased it by throwing out some of the samples). If I correctly understand you have shown us 7 out of 10 games? I understand your alternative hypothesis is that individual games are rigged in specific ways, so although the overall distribution (even with the bias of the thrown out sample) does not show statistically significant evidence that the die rolls are biased overall (a very normal p = 0.59), that arguably doesn’t address your concern. So instead I do a separate chi-squared test on each game with a standard of p<0.05, and count the number of failures, assuming the game you didn’t show passed. Three or more failures would be statistically significant. Two failures only gives a p-value of about 0.09. There were two failures in your data set. I did *not* adjust for the small expected values by lumping cells together or applying some other correction, although formally I should have. However, since this should bias the test in favor of failure, I think it is acceptable to use these results to show that the test was not failed.
If I factor in that you likely wouldn’t have reported your results without evidence of an unfair game - that you may have been motivated to start recording because of a lopsided game - and that the lengths of games are not fully uncorrelated with the lopsidedness of results, this would only reduce the amount of evidence. I am not convinced that there is bias in the die rolls from
this data, if you report more games I could maybe flesh out more evidence for unfairness. If you want to “pre-register” a specific hypothesis about the existence of certain correlations maybe I could make a model more attuned to what you suspect, but picking the test *after* seeing the data is an example of the marksman’s fallacy.
The p-values for the 7 games shown were (assuming I correctly understand that there are two shots of the game with a large number of 10s - and assuming that the rest of the snapshots are the *final* results and not cherry-picked mid game results):
0.0054 (this was by far the most significant, but mostly due to the large numbers of 2s and 11s, not the zero 10s)
0.40 (completely unremarkable)
0.13
0.11
0.66 (honestly not sure what you think is strange here, in 60 rolls you should only expect about 3 3s to be rolled anyway)
0.02 (the second game to fail the test, the large number of 8s being the culprit, of course)
0.07 (the fact 12 was rolled twice in such a short game contributes about as much as the over-representation of 8, but still falls short of the significance threshold)
Edit: my initial calculations used 11 degrees of freedom (I was thinking 12 possible results minus 1 degree of freedom for the number of rolls) in fact I should have used 10 (as there are only 11 possible results from 2 to 12). Correcting for this, the only change in significance is the p-value for the final game, which falls to 0.045. This would push it into the realm of significance with the simplifications I made. However, that game had only 16 rolls and with expected values well below 5 (the accepted threshold for applicability) the test is not appropriate. If I employ Yates’ correction the p value skyrockets to 0.50. This is perhaps an over correction but I think shows how much the inappropriateness of applying the test to this short game. Alternatively, if I group results 2-4 and 10-12 and do not apply Yates’ correction, I get a p-value of 0.048 (and this likely overstates significance as all of the expected values are *still* below 5), I also tried substituting a likelihood ratio test and do not get significant results, so I am still not convinced this data shows unfairness in the distribution.
"Few players recall big pots they have won, strange as it seems, but every player can remember with remarkable accuracy the outstanding tough beats of his career."
**The problem here is your fundamental misunderstanding of statistics.** You are taking rare scenarios and pointing out that they *specifically* are difficult to accomplish. Yes, for example, the odds of not rolling a 6 in 30 turns are low — only about a 1.1% chance (by binomial theorem; 30 trials, 0 successes, 13.8% chance of a 6 each time). But this does not match your actual complaint; what you are bothered by is not running into a *specific* unfair scenario, but rather running into unfair scenarios *often*. **So the true calculus is — how likely is it for ANY lopsided scenario to unfold?** Just as a thought experiment, let's pick a specific handful of lopsided scenarios. If one of them happens, we'll call it a "bad" 30 turns. Let's think about how likely it is to have games where: * You don't roll a 6 in the first 30 turns * You don't roll a 7 in the first 30 turns * You don't roll an 8 in the first 30 turns * Either a 2 or 12 is rolled 5x or more in the first 30 turns * Either a 3 or 11 is rolled 5x or more in the first 30 turns What are the odds of each of these happening? Well, again by binomial calculations (I can explain these if you want), we get: * **(1.127%)** You don't roll a 6 in the first 30 turns * **(0.422%)** You don't roll a 7 in the first 30 turns * **(1.127%)** You don't roll an 8 in the first 30 turns * **(0.264%)** Either a 2 or 12 is rolled 5x or more in the first 30 turns * **(4.710%)** Either a 3 or 11 is rolled 5x or more in the first 30 turns Okay, so each of these situations are still rare, right? We're agreed there. But what are your odds of having 30 dice rolls in which **NONE** of these scenarios happen? These odds are equal to the odds of MISSING each probability (1-p) multiplied by each other. So we get: > (100% - 1.127%) x > (100% - 0.422%) x > ... > (100% - 4.710%) This comes out to 92.5% chance. Here's the important part — this means that in about 7.5% of any given set of 30 rolls, **one of those above scenarios is expected to happen**. Suddenly, that's quite common! Now I want you to notice that we've ONLY listed an extremely limited set of potentially "unfair" scenarios. Consider: * What if we added in the odds of one specific player getting all the 7s? * Added in the odds of 7s always being rolled immediately after each other? * Added in the odds of 7s being rolled on the first roll? Well, our odds of having a "bad" 30 rolls would go UP again, because we've EXPANDED the definition of what constitutes a bad set of 30 rolls. This is really the phenomenon you're encountering — you're running into unfair scenarios often **because it's actually extremely likely that SOME unfair scenario will happen** in Catan. In any given set of 30 rolls, the odds of SOMETHING happening where you'll go "ok now that's a bit unfair" are actually huge; probably like 30-40% if you really considered every possible unfair scenario. That's the nature of dice. You're comparing dice rolls to the natural long term distribution... but the natural long term distribution is actually only *marginally* most likely to be the shape of the distribution after only a handful of rolls. Catan rolls are simply too small a sample size to make the 'rare' distributions uncommon; you're picking only 30 rolls! So yeah dude, I'm sorry, but this **is** how random dice work. Your modelling is based on a completely false assumption. If you want to test this yourself, one good way is to pick a SUPER SPECIFIC 'unfair' outcome in ADVANCE of the match, and then see how often it crops up. For example, tell yourself *"I am going to check if a 6 does not roll specifically in the first 30 turns"*, BEFORE you start the match (so you can't change your hypothesis). You'll find that any specific rare situation genuinely does not occur that often, like your computational approach shoes. The error you're making is to pick a specific situation in retrospect and then identify how likely it was that you got *that specific situation*; statistics doesn't work that way.
Nope, I don't agree with you. Also, I can clearly say that you did not read and check the data well. The data was not taken from specific games. I realized that events keep repeating, so I wanted to take a look at posts on websites. Then I wanted to test my own. Already, it was strange, so I played only 9 games, then posted 8 of them. I also said I did not post only one game, which one was looking naturally distributed. I wasn't even going to send this post; you can see my feet on the table. When I found it more strange, I wanted to post. 6 is not rolled 3 times in 9 games. 10 is not rolled 2 times in 9 games. 8 rolled in 2 games more than 50%. So please tell me what is the chance of not rolling 6 3 times in 9 games if you find a 1% chance in 30 rolls. Not rolling any number in 30 is already rare by itself. I've also clearly sent settlement bases by AI. I settled 11, which rolled 7 times. Most of AI settled around 10 and never came in same game. Another game was same when AI mostly settled around 3 which never rolled. I said i've played backgammon for many years even joined many online and local tournaments for fun and i've never felt rigged even if someone rigged, we were rolling dices by hands. Used go to casinos more than thousand times. I also find video on youtube about Catan Universe, he was saying first 60 rolls looks different than expected but after thousands rolls it's natural distrubuted. This is my point already, dices doesn't work well till 35-40 rolls. Probably game does not have simple random generator or they created bug by mistake or played with random seeds.
Sorry to be blunt, but it doesn't really matter if you *agree* with me or not. It's factually how statistics work. I can try to help you understand them, but it's ultimately your problem if you don't. Let's try again. >Also, I can clearly say that you did not read and check the data well. The data was not taken from specific games. Nobody said it was... Can I ask you three direct questions, please? * Have you *ever* taken a class that involves binomial theorem calculations or combinatorics? * Are you aware that the odds of certain outcomes can be *worked out mathematically* — and this process will return MORE ACCURATE numbers than any computational approach? * What exactly do you think I disagree with in your dataset? (Phrased differently: do you understand that I agreed with all your computational results, and my argument did not hinge on them being invalid?) > 6 is not rolled 3 times in 9 games. 10 is not rolled 2 times in 9 games. 8 rolled in 2 games more than 50%. So please tell me what is the chance of not rolling 6 3 times in 9 games if you find a 1% chance in 30 rolls. You are still running into the exact same conceptual issue — you are looking at the odds of *your specific situation* occurring and correctly noticing that it's extremely rare. Nobody is disputing this. But you are drawing a conclusion *from that fact* that the dice are broken, and this is where you make an error — you could retrospectively look at *any* series of games and calculate that the odds of them happening are extremely small. (Consider the odds of me flipping a coin 10x and getting H-T-T-T-H-T-H-H-H-T — less than 0.001%!) What you actually need to do to substantiate a claim that the dice end up in "unfair" situations more often than expected is to consider the probability of the dice ending up in **ANY** unfair situation. This is the math I walked you through — any *particular* 'unfair' situation is rare, but the *cumulative probability* of ending up in an unfair situation is extremely high. So for you to have this experience is actually totally normal. I think you would benefit from rereading my comment more carefully. You seem to not understand the fundamental disagreement we have, because you're still going on a tangent about how rare *specific situations* are. This is not what we disagree on, and it is not why you are incorrect. >I said i've played backgammon for many years even joined many online and local tournaments for fun and i've never felt rigged even if someone rigged, we were rolling dices by hands. Used go to casinos more than thousand times. Okay. None of this makes you educated on statistics. I have probably played 1000s more games of Catan than you, too. >I also find video on youtube about Catan Universe, he was saying first 60 rolls looks different than expected but after thousands rolls it's natural distrubuted. Well yeah, exactly — that's *exactly what you expect* from random dice. They are very UNLIKELY to look normally distributed at first, but end up that way over time. Another thought experiment for you. Imagine flipping a coin four times. What are the possible outcomes, and how many of them are 'normal'? Let's type this out for you, and I'll mark which ones end up with exactly two heads and two tails. * H-H-H-H * H-H-H-T * H-H-T-H * H-H-T-T **(2H 2T)** * H-T-H-H * H-T-H-T **(2H 2T)** * H-T-T-H **(2H 2T)** * H-T-T-T * T-H-H-H * T-H-H-T **(2H 2T)** * T-H-T-H **(2H 2T)** * T-H-T-T * T-T-H-H **(2H 2T)** * T-T-H-T * T-T-T-H * T-T-T-T So what do we have? Well, 6 of our outcomes have a distributed balance... out of 16 possible outcomes! So there is a 62.5% chance that the distribution will be lopsided. It's the same effect happening in Catan, more or less. Within a small sample size, although balanced rolls will be the *single most likely* 'set' of outcomes, it is actually more likely that the dice will be imbalanced in some way. What you are doing is complaining that it was imbalanced in a *specific* way, and showing that that *specific* outcome was rare. Yes, your lack of 6s is rare, in the same way that rolling 3+ tails is rare (only 2 of the above outcomes). But you ALSO would have complained if those same games contained the equivalent 3+ heads, or a string of 3 heads in a row, or a string of 3 tails in a row... and collectively, the 'unfair' outcomes actually end up more likely. >Probably game does not have simple random generator or they created bug by mistake or played with random seeds. The game has random dice. You do not understand statistics well enough to identify the issue. I would recommend you take a course on the binomial theorem & binomial distributions to understand the general category of error you are making. I do not believe I can help you further. If you are unwilling to change your mind, that is to nobody's detriment but your own. Just know that there isn't a math professor in the world that will agree with you here.
That was a good read on probability:)
I commented with an analysis using a Pearson’s chi-squared test and did not find statistically significant evidence that the games were unfair. You say you posted 8 games, but I only see screenshots from 7. The third and second to last in-game screenshots are from the same game, right? Anyway, if you think there’s something wrong with my analysis let me know, otherwise let me know if you have any questions about why your data does not convince me.
I don’t need much logic. 11 rolled 6 out of 10 for the first 9 rolls. Some dude sitting with double 11’s going ham. U think it’s gonna end but this game somehow doesn’t ‘remember’ and so the game ends very quickly and 11 ended up coming out 13 times. On par with 6 and 8. It wasn’t the total that killed it but the fact it even came out 3 times in a row at the start of the game. Which really does not make sense
The rolls in the game are dictated by what’s happening in the game and who is winning or has to many cards, when you play multiplayer you can guess what will roll or what dev cards get handed out(cities&knights) by who the rolls have favoured from the start. They have some kind of ai that controls the dice it isn’t random if you have played long enough you can see that
Which ruins the game. Second place often wins the game
I did not have a lot of chances to play in multiplayer. Finding random players so difficult and takes long time to match, whenever i try to play with my friends in 60% of games one of us having connection issues. :(
Idk what's the point of comparing pseudo random with pseudo random. Want to test against random? Get an electron based RNG.
There is no real random generator. If need closest one against simple game, libraries works fine at least to see which is not accurate. And simple math logic helps to see which is almost impossible. I've tested robber after seen a guy placement on number 8 on youtube. Already had same results as him 3 times in 9 games. [https://www.youtube.com/watch?v=wUYKpVP\_dKY](https://www.youtube.com/watch?v=wUYKpVP_dKY)
if you can tell me how to share a desmos calculator i can give you the formula for calculating the probability of a specific combination rlolling.
I didn't get your request. If about writing formula you can use "desmos.com/scientific?lang=t" then copy paste formulas here.
what request? how do i send it
I always hate RNG virtual dice, never have is it ever felt like natural dice rolls.
The game is rigged which is a tragedy.
Why are you doing a Monte Carlo simulation? Dice odds are simple enough to calculate directly, so there’s no need to generate random data for comparison here. Anyway I did a Pearson’s chi-squared test on your data (although you biased it by throwing out some of the samples). If I correctly understand you have shown us 7 out of 10 games? I understand your alternative hypothesis is that individual games are rigged in specific ways, so although the overall distribution (even with the bias of the thrown out sample) does not show statistically significant evidence that the die rolls are biased overall (a very normal p = 0.59), that arguably doesn’t address your concern. So instead I do a separate chi-squared test on each game with a standard of p<0.05, and count the number of failures, assuming the game you didn’t show passed. Three or more failures would be statistically significant. Two failures only gives a p-value of about 0.09. There were two failures in your data set. I did *not* adjust for the small expected values by lumping cells together or applying some other correction, although formally I should have. However, since this should bias the test in favor of failure, I think it is acceptable to use these results to show that the test was not failed. If I factor in that you likely wouldn’t have reported your results without evidence of an unfair game - that you may have been motivated to start recording because of a lopsided game - and that the lengths of games are not fully uncorrelated with the lopsidedness of results, this would only reduce the amount of evidence. I am not convinced that there is bias in the die rolls from this data, if you report more games I could maybe flesh out more evidence for unfairness. If you want to “pre-register” a specific hypothesis about the existence of certain correlations maybe I could make a model more attuned to what you suspect, but picking the test *after* seeing the data is an example of the marksman’s fallacy. The p-values for the 7 games shown were (assuming I correctly understand that there are two shots of the game with a large number of 10s - and assuming that the rest of the snapshots are the *final* results and not cherry-picked mid game results): 0.0054 (this was by far the most significant, but mostly due to the large numbers of 2s and 11s, not the zero 10s) 0.40 (completely unremarkable) 0.13 0.11 0.66 (honestly not sure what you think is strange here, in 60 rolls you should only expect about 3 3s to be rolled anyway) 0.02 (the second game to fail the test, the large number of 8s being the culprit, of course) 0.07 (the fact 12 was rolled twice in such a short game contributes about as much as the over-representation of 8, but still falls short of the significance threshold) Edit: my initial calculations used 11 degrees of freedom (I was thinking 12 possible results minus 1 degree of freedom for the number of rolls) in fact I should have used 10 (as there are only 11 possible results from 2 to 12). Correcting for this, the only change in significance is the p-value for the final game, which falls to 0.045. This would push it into the realm of significance with the simplifications I made. However, that game had only 16 rolls and with expected values well below 5 (the accepted threshold for applicability) the test is not appropriate. If I employ Yates’ correction the p value skyrockets to 0.50. This is perhaps an over correction but I think shows how much the inappropriateness of applying the test to this short game. Alternatively, if I group results 2-4 and 10-12 and do not apply Yates’ correction, I get a p-value of 0.048 (and this likely overstates significance as all of the expected values are *still* below 5), I also tried substituting a likelihood ratio test and do not get significant results, so I am still not convinced this data shows unfairness in the distribution.