Quote:
Originally Posted by Klyka
I will assume that by a set, you mean a PP that hits the board. That is, we exclude AK that hits a paired board.
There are 6*11 combos of pocket pairs, and 4 combos of AK, for a total of 70 combos. So, the chances of player #3 holding a pair is 66/70.
When he holds a pair, he has 2 outs from the remaining 46 cards, so the chance he at least hits one of them is 1 - (44/46)^5.
So the combined probability of him (1) holding a pair and (2) hitting a set with it, is 66/70 * (1 - (44/46)^5) ~ 18.8%
If you want to account for trips as well, the probability is slightly higher.
EDIT: Klyka has already corrected the above in a subsequent post.
This is incorrect. The probability of missing is 44/46 only on first card of the board, not every card, so we cannot simply raise this to the 5th power. The probability of missing on the 2nd card after missing on the 1st card is 43/45, and the probability of missing on the 3rd card after missing on the first 2 is 42/44. The OP was interested only in the flop, so the correct probability is
66/70 * [1 - (44/46 * 43/45 * 42/44) ]
=~ 12.0%
We can do this with combinations as
66/70 * [1 - C(44,3)/ C(46,3) ]
=~ 12.0%
where C(46,3) is the total number of flops, and C(44,3) is the number of flops which do not make a set. Note that this 12.0% will also include quads and full houses.
Last edited by BruceZ; 02-02-2008 at 11:40 AM.