Odds of three people at a 9 ring game having the same hole cards

Question

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.

Answer 1

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).

Answer 2

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

Answer 3

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