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I was just wondering what the odds are of three people at a 9 ring game having the same hole cards. Played a hand today online where the pot split 3 ways and each player held j9. I don't remember ever seeing this happen and it was right after a few suspect hands where full houses were beat by quads.

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  • This is some exciting (and relatively useless) piece of math. Basically the chances are very low. If there were only 3 people they are - 0.00002404889 = 0.0002% If I find out an easy calculation for 9 I will write an answer with an explanation
    – Daniel
    Apr 24 '15 at 14:14
  • Never thought about this but i've played many hands that ended with all of us having some kind of Ax variation, so it can't be that uncommon. This variation seems more useful than try to find the exact hole cards.
    – user1165
    Apr 24 '15 at 21:10
  • haha speaking of full houses being beaten by quads, yesterday i started at a (online, play money) table with AA on the button, wound up tripling my stack. then during the course of about half an hour i saw quads beat a full house four times, one time of which I was on the winning end of and once on the losing end.
    – user1934
    Mar 31 '17 at 21:13
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Unfortunately, this is the kind of probability question that would take a couple of hours and reams of paper notes to get the exact answer. But simulating it is easy (I have loads of C code for that).

I got about 1 in 500. (205423 hands out of 100000000).

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This is for just 3 players on the table

First player has to be unpaired
94%

Second player has nine ways to match
0.73%

Third player has four ways to match
0.35%

Total
0.0025% = 1 / 40782

I think you would multiple by combin(9,3) = 84
And this is pretty close the the simulation from Lee 1 / 485.5

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i ran a histogram on the number of different hands in a 9 player game. this is very preliminary, but i got the results:

1000000 5.0<=8.833096/1000000<=9.0 0,[0,0,0,0,0,3,212,9030,148196,842559],0 NaNs: 0

so 84% of the time all the hands are different. 15% of the time two hands are alike. 1% of the time 3 hands are alike or two sets of players have the same hand or some other combination reduced the number of different hands.

edit: larger sample:

1000000000 4.0<=8.833342/1000000000<=9.0 0,[0,0,0,0,4,2238,220758,8874325,148238607,842664068],0 NaNs: 0
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