December 16th, 2018 at 7:31:12 AM
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I've been puzzling over how to most quickly detect an unfair/unrealistic deck in online holdem. The decks seem to produce unusually good hands, and in particular "dramatic betting situations" where two people have a flush or a straight. In some cases, you can detect this in 100 hands...but in others it might take 100K hands before you're really sure.

But maybe I have an answer, and here's where I need the math help.

The standard tables will show you the probability of your getting a given hand--a single set of 7 cards of which you pick the best 5. So let's look at four of a kind: the likelihood is ~0.2%. Therefore, the likelihood of my *not* getting that hand is 99.8%.

So, if there are 9 players, isn't the likelihood of *all of us not getting that hand* 98.2% (that is, 0.998 ^ 9)?

If that logic is correct, let's take everything that beats a flush: the likelihood of my getting one of those hands is 2.8%, so in 9 hands the likelihood somebody gets one of those hands is 15% (that is, 1 - 0.982 ^ 9), assuming all 9 hands stay in until the end. Virtually all that 15% is due to the full house...and unless the players are nuts, the real world would never be the full 15% because players would drop out due to high bets early on.

If all that's correct, the fair-deck detector can simply be "if everything better than a flush comes up more than 12% of the time, it's a 'juiced' deck."

If all that's correct, the surprise (to me) is how often 3 of a kind or better will show up: 78% of the time, assuming 9 hands stay in through the river.

Comments? Corrections?

But maybe I have an answer, and here's where I need the math help.

The standard tables will show you the probability of your getting a given hand--a single set of 7 cards of which you pick the best 5. So let's look at four of a kind: the likelihood is ~0.2%. Therefore, the likelihood of my *not* getting that hand is 99.8%.

So, if there are 9 players, isn't the likelihood of *all of us not getting that hand* 98.2% (that is, 0.998 ^ 9)?

If that logic is correct, let's take everything that beats a flush: the likelihood of my getting one of those hands is 2.8%, so in 9 hands the likelihood somebody gets one of those hands is 15% (that is, 1 - 0.982 ^ 9), assuming all 9 hands stay in until the end. Virtually all that 15% is due to the full house...and unless the players are nuts, the real world would never be the full 15% because players would drop out due to high bets early on.

If all that's correct, the fair-deck detector can simply be "if everything better than a flush comes up more than 12% of the time, it's a 'juiced' deck."

If all that's correct, the surprise (to me) is how often 3 of a kind or better will show up: 78% of the time, assuming 9 hands stay in through the river.

Comments? Corrections?

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