Odds someone flopped a set

[Waterfall]

This has probably been discussed here before but I couldnt find an answer. What are the odds that on a 9 handed table someone has a set vs your top pair or overpair? I know the odds of hitting a set yourself are 7.5:1 but how often can you expect someone to hit one against you?

[Copernicus]

not counting trips on a paired board?
Roughly the answer should be something like .0045 x 12 / 8.5 x number of players. .0045 is the probability of a specific pair, 12 eliminates the rank of your overpair/top pair, which cant also be a set, 1/8.5 is the probability of the pair matching the board.

So assuming a 9 player game (8 opponents)the chances are roughly 1 in 20. Obviously preflop action might change that..eg an early position raise is less likely to have the lower pairs, a limper is less likely to have the higher pairs, etc.

[BruceZ]

[ QUOTE ]
This has probably been discussed here before but I couldnt find an answer. What are the odds that on a 9 handed table someone has a set vs your top pair or overpair? I know the odds of hitting a set yourself are 7.5:1 but how often can you expect someone to hit one against you?

[/ QUOTE ]

The probability that at least 1 of 8 opponents was dealt a pocket pair which would have flopped a set when there are 3 different ranks on the flop and you hold an overpair can be computed exactly with the inclusion-exclusion principle. There are 3 pairs that make each of the 3 sets, so the probability of a particular player having one is 9/C(47,2). The probability that at least 1 player has one is:

8*9/C(47,2) -
C(8,2)*9*6/C(47,2)/C(45,2) +
C(8,3)*9*6*3/C(47,2)/C(45,2)/C(43,2)

=~ 6.5% or 14.3-to-1.


If you hold top pair, then there are only 7 ways to hold a pocket pair that makes a set, and the probability that at least 1 of 8 players has one is:

8*7/C(47,2) -
C(8,2)*(6*4 + 1*6)/C(47,2)/C(45,2) +
C(8,3)*[6*(3*1 + 1*3) + 1*6*3]/C(47,2)/C(45,2)/C(43,2)

=~ 5.1% or 18.6-to-1.


Note that these are the odds that someone would have made a set if they stayed to see the flop. Since not all pairs see the flop, it should happen to you somewhat less often than this.

[Copernicus]

BruceZ (or was it Aaron) thinks that approximations imply mediocrity...thats the difference between us applied mathies vs the theoretical ones. The difference between 19:1 and 18.6:1 is irrelevant in the real world, and you can compute the 19:1 yourself with a little thought and a few basic probabilities. [img]/images/graemlins/smile.gif[/img]

In fact us applied guys never accept a technically derived answer without some seat of the pants approximating to verify the reasonableness of the answer.