four of a kind odds

[groo]

In a hold-em hand where there are 3 same cards up, what are the odds of hitting four of a kind? And what are the odds of hitting the four of kind in that manner 5 times in 1000 hands? Just curious, since my full house has lost to that four of a kind 5 times in the last 1000 hands, and Ii find it difficult to believe the odds where against me. Thanks

[gaming_mouse]

groo,

The question you asked actually isn't specific enough to answer. The first thing to consider is what the betting was like on the flop and turn. If someone is going to the cap with you, having that fourth card becomes more likely for them. However, figuring the effects of betting into a probability calculation is not an exact science.

Having said that, you CAN calculate the probability that the remaining quad card is in one of your opponent's hands, if you assume that your opponent's are dealt random hands out of the remaining deck (This assumption is false, but it will at least provide us with a rough estimate).

For example, if there are 3 board cards of the same rank after the river has been dealt, that means there are 45 unseen cards in the deck, with 1 remaining quad card. If you are heads up with one remaining opponent, the chance that he has that final card is:

1 - (44/45)(43/44) = .044, about 4 and a half percent.

Note, however, that if there are 3 high cards on board (like queens or aces), this number is actually higher, because people are more likely to play hands containing high cards. And if you are up against more than one opponent on the river, it will be higher still.

In any case, 5 in a 1000 is only one half of one percent, so even if you were heads up on the river, you should not be surprised that you were beaten as often as you were. In fact, you were quite lucky (you should have been beaten 44 times).

There are two reasons your number is so much less 44. First, sometimes when your full house was beaten by quads, the oppo was holding a pocket pair, whereas our calculation assumed 3 same-rank cards on the board. Second, when people see someone betting on a board with three same-rank cards, they suspect a bluff and will call down with things like A high. This inflates the number of times you will win.

gm

[groo]

Thank for your reply.