hitting a full house after the flop

[philopker]

Let's say you hit a set on the flop with your pocket pair but there is a flush or straight draw out there. What is the probability that your set will improve to full house or quads by pairing the board by the river and taking care of the draws? I want to say its roughly 40% but I could be terribly wrong.

[BruceZ]

[ QUOTE ]
Let's say you hit a set on the flop with your pocket pair but there is a flush or straight draw out there. What is the probability that your set will improve to full house or quads by pairing the board by the river and taking care of the draws? I want to say its roughly 40% but I could be terribly wrong.

[/ QUOTE ]

There are 7 cards that can make you a boat or quads on the turn, and if those don't come, there are 10 that can make a boat or quads on the river. The usual calculation assumes that you don't know anything about your opponent's cards, in which case there would be 47 unseen cards, and the probability would be:

7/47 + 40/47 * 10/46 = 33.4%.

or you can compute it as:

1 - 40/47 * 36/46 = 33.4%.

However, if you know that your opponent has a flush or straight draw, then you know that at least one of his cards is not one of the cards you need, so with 46 unseen cards, the probability becomes:

1 - 39/46 * 35/45 = 34.1%.

That assumes that his other card is completely unknown, but of course not all cards can give him a flush or straight draw. The most likely scenario is that he doesn't hold any of the cards you need, and in that best case scenario the probability becomes 34.7%. If he does hold one of the cards you need, it drops to 30.4%.

[thejameser]

for practical purposes 1/3 chance, for technical see the main man's analysis.