statistics help please

I am looking to calculate the odds of a run of bad luck being due to chance. E.g at the poker table given say a 100 hands and the dealt cards, each hand before the flop (e.g AA vs KQ) will have a different probability of the player winning at the river (ignoring skill here). So say the list goes hand A, 60 chance of winning, actually won, hand B 90 chance of winning, actually lost, etc - what test would I have to perform to determine if the game was rigged? How many hands would I have to test?

I am not familiar enough with poker odds to try to calculate something for the situation you describe, but here’s what you do in general. Let’s take something simple, like you’re trying to determine if a coin is fair.

The chance of hitting heads is 0.5 each flip. The chance of hitting heads twice in a row with a fair coin is 0.5[sup]2[/sup], or 0.25. Each additional flip adds 1 to the exponent.

If you flip four times, there is a chance of 6.25% that you get heads four times. I believe this is equivalent to saying that there is a 6.25% chance that the coin is fair, if you allow for the possibility of an unfair coin biased to give heads every time.

So now the question becomes subjective. How low do you need that percentage to go before you are convinced that the coin is actually unfair? I think most of us would not jump to a conclusion of an unfair coin after only 4 flips.

Same thing for poker.

An article on how to check if a coin is fair or not. Be warned it is way way more complicated than the method proposed by CookingWithGas.

Great question. Some thoughts.

(1) Very hard to ignore the other elements of poker. What do the other players have, how many players were in the hand, how do they play, how much was the pot worth, etc. With only 100 hands, there’s going to be many outliers.
(2) In a previous poker thread, someone had a great link. (Sorry I can’t find it.) One poker site published their historical stats over millions of hands played on that site, so you could look up that AA won 77.1% of the time.
(3) You are confusing a few different hypothesis. Statistics can help show that all else being equal, your outcome was likely or unlikely. If it was extremely unlikely, you assume that something else is up. There is no way to determine if the something else is skill, cheating, or other factors.
(4) I’m not sure, but this feels like a modified form of the coin-flipping test. I would look at the Binomial Test. I’m not sure how to deal with each hand having a different probability. Perhaps (a bit crudely) you could divide the hands up into poor, medium, and good starting hands and analyze each separately with the average probability of the hands in that bucket.
(5) Cooking’s test is not relevant here. You probably didn’t lose every single hand, you only lost some more than you expected. Imagine you flipped a coin 100 times and it came up heads 41 times, what should you conclude? That is more relevant. Or you rolled a die 36,000 times and it came up 4 only 4,200 time. That is what the binomial theorem is for.
(6) The conclusion is subjective, and it isn’t. The “answer” in statistics is the p-value. Generally speaking, if the p-value is below 5% you assume your theory was right (that there’s more than luck going on). But you may feel you don’t need evidence as strong, or may require more proof.
(7) Asking how many hands are needed to draw a conclusion is equivalent to asking about the power of the test. You can look up the power of the binomial test if that’s what you end up using.

How many people are at the table? The odds change dramatically with 10 people compared to heads-up. Heres a calculator. Put in the number of players and any hands you chose. It will do the work for you.

The other elements of poker make it extremely difficult to apply standard tests here. There is a certain amount of luck involved in poker, but it’s first and foremost a game of strategy.

thanks for the replies - muttrox came closest I think with the main problem - that each hand has a different odds of winning so the simple binomial (e.g. fair coin test) will not do. I was well aware of the poker hand calculators - I was going to use those to calculate the probability of each hand.

One problem is that most of one’s wins or losses tend to be on a few hands where one goes all in, so it would be easy to manipulate just a few key hands and still pass any simple statistical test - e.g. the numbers of aces or pairs one gets dealt.

But ignoring the poker aspects for the moment and put it simply as a maths problem- given a set of N trials and P outcomes, where the probability of success varies at each trial but is known, can one calculate the odds of a particular result (e.g. 50 wins 30 losses) being due to chance?

You could, even if you’d need some heavy duty computations. The number would be pure bullshit due to your ignoring the actual complexity of the problem, though.

Holy crap. I will never try to answer a statistics thread again :smack:

bullshit will do ! - any links or a name of the method?