Two Players Having Identical Hands
In watching some poker tournaments on TV, I was a bit surprised by how often two players have the same hand.
You can show theoretically that heads-up, it happens a bit less than once in a hundred hands (16/17 * 9 +1/17*1)/1225, so you will occasionally see it but not very often. However, with a table of more players, there are more possibilities.
For example, with 6 players, there are 6*5/2 =15 player pairings, thus more chances for two having the same hand.
So, I went about trying to model that situation. After thinking about it for a half minute or so, no way could I solve it without considerable effort, if at all. So, I’m in the midst of writing a simulation program and not being much of a programmer it’s been a bit of a struggle, but I am sure I will come to some completion.
It would be great if someone can do the math so I can have a check on my effort as well as answer this critical question that at least one other poker math nerd has thought about.
Oh, as an aside. I also wrote this thread to ensure that the probability forum has at least one new posting a month. Hopefully there are a lot more.
3 Replies
I started trying to work it out but I think there are so many cases that need to be worked out that it's very cumbersome.
I ran a simple Monte Carlo simulation with 2 million trials, and I found a probability of 10.09%.
I know you were asking for a mathematical solution, but hopefully my simulated results can help you to confirm your own results as well 😀
For a 9 handed game, it looks like the probability of two players sharing the same hand would be 22.84%.
I'm kind of surprised the probabilities are that high, but it now makes more sense when you mentioned that you've seen its quite common to see in streamed tournaments.
EDIT: Also funny enough, but the probabilities seem to scale almost linearly with the amount of player pairings. Heads up has one player pairing while 6 handed has 15 player pairings, and the probability is 14.62x higher.
Same with 9 handed. There are 36 player pairings and the probability is 33.1x higher.
I got exactly the same results, so we're either both right or both wrong 😉. But, since you only have 8 postings as of this writing, I don't know if I should trust you 😀 😀
