Running a statistical test / Forming a hypothesis

[Homer]

Can someone explain how one goes about doing this? Often, someone on the internet gambling will claim that Party Poker is rigged because their flush draws don't come in often enough, more of a certain card comes on the turn and river than is expected, etc. I would like to prove them wrong statistically.

[jason1990]

One thing you could do is simply compute the fraction of flush draws that hit, compute a confidence interval about this value, and then confirm that the probability of a flush draw hitting (in a fair game) is in this interval.

[Homer]

[ QUOTE ]
One thing you could do is simply compute the fraction of flush draws that hit, compute a confidence interval about this value, and then confirm that the probability of a flush draw hitting (in a fair game) is in this interval.

[/ QUOTE ]

Thanks, Jason. I sort of had a vague idea that this would be the general process. However, I'm not sure how to do all of the math.

-- Thanks, Homer

[TomCollins]

I'm really surprised you wouldn't know this with all the math on your webpage.

I'll do a simple example. Suppose you have a coin that you believe is not fair. If it was fair, it would heads 50% of the time, and tails 50% of the time. Of course, since you cannot do infinite trials, you will have to do a limited set of trials. Suppose you flip it 100 times. Obviously this sample size is pretty bad. It may end up heads 52 times and still be a fair coin. But suppose it was heads 60 times. Is this coin fair?

At first glance, it seems as if it is not. So this is where the test comes into play. The distribution of a random coin should match a binomial distribution, which has tables. These tables (which can be calculated as well), will tell you how many trials out of a fair distribution will score above (or below) a certain point. The binomial distribution looks very similar to a normal curve for large sample sizes.

Binomial Distribution (http://www.stat.berkeley.edu/~stark/Java/BinHist.htm)

Plug in the parameters here, and there is a 1.8% chance that a fair coin will score 60 or more heads. So this test fails. In other words, we cannot prove the coin is not fair. However, this does NOT prove the coin is fair.

Unfortunately, you cannot really prove Party Poker is fair, you can only prove it was not fair. But, you could use these numbers to say with a confidence that a certain card does not come up with an unusual amount of times.

It's been a while since I've done this, but anything I screwed up, feel free to correct me.

[BlueBear]

I believe it's not that simple.

For example, when you draw to a flush at the turn, in calculating the probabilities the flush will fill, there is the effect of the unknown hole cards of the opponent, as the opponent is somewhat more likelier to hold another (or two) other flush cards. Eg. you hold Ad Qd, board is 5d 7d 8d 2c, and your opponent does call all the way the river, he will do so more often holding another diamond, eg. Kd. This reduces the number of flushing outs in the river.

I believe if a study is conducted to compute the fraction of flush draws that hit without making the above considerations, something will seem amiss.

Any statistic experts care to comment on my possibility faulty logic?

[Homer]

[ QUOTE ]
I'm really surprised you wouldn't know this with all the math on your webpage.

[/ QUOTE ]

I'm getting a little rusty. I did the webpage stuff a while ago, so I hoping someone would walk me through an example so I wouldn't have to go digging through my stats books.

[ QUOTE ]
I'll do a simple example. Suppose you have a coin that you believe is not fair. If it was fair, it would heads 50% of the time, and tails 50% of the time. Of course, since you cannot do infinite trials, you will have to do a limited set of trials. Suppose you flip it 100 times. Obviously this sample size is pretty bad. It may end up heads 52 times and still be a fair coin. But suppose it was heads 60 times. Is this coin fair?

At first glance, it seems as if it is not. So this is where the test comes into play. The distribution of a random coin should match a binomial distribution, which has tables. These tables (which can be calculated as well), will tell you how many trials out of a fair distribution will score above (or below) a certain point. The binomial distribution looks very similar to a normal curve for large sample sizes.

Binomial Distribution (http://www.stat.berkeley.edu/~stark/Java/BinHist.htm)

Plug in the parameters here, and there is a 1.8% chance that a fair coin will score 60 or more heads. So this test fails. In other words, we cannot prove the coin is not fair. However, this does NOT prove the coin is fair.

Unfortunately, you cannot really prove Party Poker is fair, you can only prove it was not fair. But, you could use these numbers to say with a confidence that a certain card does not come up with an unusual amount of times.

It's been a while since I've done this, but anything I screwed up, feel free to correct me.

[/ QUOTE ]

Thanks, this all looks vaguely familiar. I'll take a look at that url...

-- Regards, Homer

[BeerMoney]

[ QUOTE ]

Unfortunately, you cannot really prove Party Poker is fair, you can only prove it was not fair.

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This statement is accurate.....

I'm just wondering, what would prove that it is unfair?
Would it be unfair if you hit your flushes more/less often than expected?? See what i'm saying?

[Leo99]

Exactly. If you had a 4 flush with one card to come on the river you know the probability is 9/46. You expect to hit 9 out of every 46 times. If you hit it 1 out of 460 times or 400 out of 460 times you'd have to question the fairness of the game because your observed probability differs so much from the theoretical probability. It's very similiar to if you flipped a coin 460 times and got 450 heads and 10 tails. Is the coin fair (i.e. heads or tails has a 50/50 chance)? You start to get into the probabiltiy of false positives and false negatives.

[o0mr_bill0o]

[ QUOTE ]
I believe it's not that simple.

For example, when you draw to a flush at the turn, in calculating the probabilities the flush will fill, there is the effect of the unknown hole cards of the opponent, as the opponent is somewhat more likelier to hold another (or two) other flush cards. Eg. you hold Ad Qd, board is 5d 7d 8d 2c, and your opponent does call all the way the river, he will do so more often holding another diamond, eg. Kd. This reduces the number of flushing outs in the river.

I believe if a study is conducted to compute the fraction of flush draws that hit without making the above considerations, something will seem amiss.

Any statistic experts care to comment on my possibility faulty logic?

[/ QUOTE ]

the other players' hole cards are all still random, and thus might as well be considered part of the deck. when basing your play in an individual hand, it might be prudent to think that maybe your opponent is on the same draw, however in aggregate it should all even out, so i don't think this is a consideration you have to make.

[BlueBear]

This is true is all hands are played until the showdown. However, in reality, only hands with certain "made" or "drawing" value goes all the way to the river.

So, I think that if a hand goes all the way to the river, there will tend to be "duplication" in the drawing values between hands of different individuals. But I believe this effect cannot be neglected if somebody does endeavour to conduct this study.