If 2 people go to the casino, 1 of them is blind, the other is not.
When arriving at the table, the sighted person sees that in pockets 0 32 15 19 4 21 2 25 17 34 6 27 there are particles that could obstruct the entry of the ball, the blind man does not see that.
The 2 players decide to play and write down what happened in the roulette. The blind man will play a dozen, the 1st dozen. The other player would also play 12 numbers, but making guesses he would play the next adjacent 6 numbers on the cylinder , the 6 numbers 26 3 35 12 28 7 and 13 36 11 30 8 23(12 numbers)
By scoring and playing 1000 balls results are: the blind have tied and the other player has won. Over 1000 balls, the dozen chosen by the blind man came out 334 times, which he did not lose. And the 12 numbers chosen by the sighted player came out 361 times so they came out 28 times more than the 12/36 payment rate and 36 times more than the 12/37 average. For the blind player, he simply analyzes 1000 balls where a sector of roulette has come out based on the +2.5 standard deviations. But, for the sighted player, the sector played and what happened and we could say that the probability of choosing 12 numbers in advance and achieving +2.5 standard deviations happens by chance in 1 test of 1000 each more than 30 tests of 1000. That is to say that it has had a very strong streak, or its prediction obeys some previous subjective knowledge. There is a saying that says: "He who does not know is like he who does not see" The blind will go the other day to play another dozen or the same, the other player will apply the same prediction, and its result will have a very high probability of repeating itself.
I think this kind of example has never been raised before. We can use subjectivity, or maybe it's something from Bayes giving A and B some of the assumptions and events in the example.
What do you think?

regards
ybot

i think this has gotten quite a bit of treatment over time, at least in practice. i'm not very academically inclined so i can't cite any sources, but i think every (actual, practicing) advantage player knows exactly what you mean.

i looked through your posting history and found a thread from 2012 that you'd participated in where a highly successful advantage player tried to make a similar point to some of the resident "gambling geniuses" and they could not grasp it. he even went out of his way to construct a couple very obvious thought experiments, but the braintrust couldn't get around the X's and O's of it all, and failed to see the bigger picture. testing a hypothesis and plumbing for results are different things.

Fubster, you are right
This probability calculation must be hard to measure.

the human mind really is interesting in its pattern making.

Can you solve this problem?

I don’t quite understand what the problem is. Casually applying Baye’s, from the blind man’s perspective observing a +2.5 standard deviation result is far, far more likely to be random variance than finding a person with an edge. Also, the “particles” on the wheel is not very strong evidence that an edge could be obtained, assuming anyone can potentially see them, considering that it is likely that someone could have found and exploited the flaw already if there was one, leading to the flaw being discovered and fixed by the casino.

From the blind man it mean nothing, there are many way to gather 12 numbers from the 37 available, he does not know that the other player picked them in purpose.
This particles could mean nothing or everything. You may calculate the chance of getting this +2,5sd and a likelihood to repeat a similar performance next time.
We are making this example to go to its extremes because a blind man cannot see what w
happens.
The degree of knowlegde on a game could give you more specifict data than others.

In any coin flip game, betting the side which has come up more often will take advantage of any bias that exists, but also won't hurt you if there is none. No math necessary.

Hi newoldguy, yes, it is a way to take advantage of it.
Math helps to accelarate a decision. To know the probability of events seen

Can you calculate any probability?

Suppose you make a statement(hipotesis) that 12 numbers which must rate 12/37 would rate 14/37, and it hits 13/37 14/37 or 15/37. This probability in 1000 trials
What is the chance that the result happened randomly?

This is a pretty standard confidence interval problem using standard deviation. Post it in the Probability Forum and you'll get a number of people willing to show you the math. I don't have time right now or would take a stab at it.

I wouldn't use standard deviation here because it's a binomial distribution, so an exact probability is easy to calculate.

The probability of hitting (denominators rounded):

<= 351/1000 bets: 1/30
<= 378/1000 bets: 1/6683
<= 405/1000 bets: 1/27814008

However, these probabilities don't accurately test the hypothesis that an edge exists or you'd see casinos banning anybody who hits 378/1000 of these bets. The OP mentioned Bayes' in his title, so I'm assuming he wants to incorporate that some way, but I'm not sure how, other than to note that casinos have an interest in protecting their games, and it's extremely unlikely that a defect that would have potentially been exposed to millions of people over the course of time would be significant. Surely other people can see these "particles," and potentially take advantage of them, tipping the casino off that something is wrong?

A potential flaw that takes expertise to spot, or was introduced very recently is far more likely to be statistically meaningful as a smaller pool of people are capable of finding it.

Few players would hit 378/1000 playing 12 numbers. It is +3.7 sd.
Standard deviation rule 68.3 /95.3/99.7 helps to undestand
@Bayes:Suppose I had 2 roulette wheels to play on. I pick the one I noticed a defect(A)( let's make this theorical example)
12 numbers I picked from this wheel hit 361(B)or 350 or 370 times in 1000.
What is the probability of hitting 361/1000(B) knowing A.
Sorry if I confuse some concepts.
I am trying to infer via bayesian tools instead of frequentists

Yes, few players would hit 378/1000, but almost all of the ones that do are just lucky.

It's very hard to try to assign reasonable numbers to do the calculation using Bayes', which is why I haven't tried to. In fact any answer here is going to be subjective.

My point before was that if you noticed a defect with your naked eyes, other people could have too and taken advantage of it before you. The game still exists so it is likely not a defect that lends any sort of advantage. Applying Bayes would just tell you that neither wheel is likely to be biased. My subjective opinion is therefore that it is still about the same probability to hit 361/1000 on the wheel whether we've noticed a defect or not. Noticing a defect makes our odds only very slightly better.

Yes, could we make subjective answers?
You said : ""if you noticed a defect with your naked eyes, other people could have too and taken advantage of it before you""
It takes an expert to notice what others don't, and, each realm have their own details to pay attention.
A regular player could not notice dealer throws clockwise and counterclockwise directions alternately, that there cold be 2 3 or 4 different balls using, many other conditions we ignore.
You may hit 361/1000 playing in any wheel. Does the player who have not make the the hipotesis need more data to be confident of a way to play?
As you said at the end ""Noticing a defect makes our odds only very slightly better."" How much better?

IRT banning said player, if casino management saw a substantial reduction in hold on this particular wheel for a substantial length of time, and they could identify said player, I am inclined to believe that player would be very closely monitored and subsequently banned if his winning pattern continued.

Of course "substantial length of time" is subjective. It is also relative to the amount paid out to said player. If the normal hold on that wheel is $10k/week, and since his play the hold has gone to zero for 3 months, then that should be enough to get him banned.

Pokerhero, it is not easy to know when to ban at this scenario. Pro players are below the radar.
We want to know the math behind?

OP - I'm wondering if you are just interested in the math, or if you are really thinking that this is something you want to do. I know you have arbitrarily picked 1000 spins - but let's suppose you do that. Estimates I've found on line are that you get about 40 spins per hour. So you could track this for 25 hours, without any reasonable expectation that you would find a biased wheel. Since the default expectation for any wheel is that there is no bias, you would have to do an awful lot of incredibly tedious work to ever find a wheel that might have a bias, and it would have to have enough of a bias to actually give an edge over the house.

I'm not saying it can't be done, because it has been documented that it has been done. I'm just thinking that it is unlikely that you will ever beat the game.

If you are just looking at how to do the math - you might want to look at a chi-squared goodness of fit test, comparing your observed frequency against the expected frequency - but instead of using 12/37 as the expected frequency use 12.33/37 - because that is the percentage you need to exceed in order to make a profit. If you use 12/37 you could find a biased wheel that still ends up with you losing money.

VBaces, in the example , the player picked these 12 numbers because he would probably know something that others didn't. He does not take data from any wheel
40 to 50 spin an hours is a regular rate. Ypu may find 60 or more an hour at online live dealer's wheel.
Chi square test takes at least 3k to give us more accurate results.
I am trying to compare someone who made a prior hipotesis against any other who didn't

Don't know where you came up with chi square needing at least 3K. If that were true, then any statistical test you use is going to need that sample size. Even so, he doesn't know something others don't know - he simply observes something that he thinks might have an impact and now wants to test the hypothesis that the results are different from random. What other people do is irrelevant.

So, if you actually have a wheel that you have observed, and think it might be biased, that is one thing. If you are just thinking that it might happen some time, that is something completely different. In either event, even if you can get 60 spins per hour, you are still looking at an awful lot of time to measure.

Also, I suggest you forget about the other (blind) person. It is very likely that if the wheel is biased in favor of your 12 numbers, a person selecting randomly (and therefore choosing a different 12 numbers) will get winning results that are significantly less than would be expected by chance. If your numbers come out more often than random, that means the other 25 numbers have to come out less often than random. If you did both and found the random group to come exactly as expected and the special numbers coming out more often - you know that one of those conclusions (biased vs not biased) has to be incorrect.

Unfortunately I know more than enough on how casinos ban players in scenarios such as this.

@pokerhero, we could talk about how land and online casinos out of public forum.

@VBAces, chi square test for 36 degrees of freedom takes some amount of date. Sorry if I said 3k was the propper trials to make a good analisis, I can be wrong.
What I am trying to find out is to detect an advantage earlier using prior physics skills or taking advantage of a flaw seen by very few people.
Volatility creates noise which makes our work harder.

You don't need 36 degrees of freedom unless you are looking at each individual number. I thought you were looking at the group of 12 numbers, and comparing the percentage that fall into that group of 12 against a known percentage of what you would expect to fall into that group (or actually against a break-even percentage).

As there are dozens of ways to make 12-number-sections you must count each number separately