calculating holdem probability - flush draw vs two-pair

[davidw]

It's been a while since I took probability... How do you combine the odds in this example:

I have two pair on the flop, my opponent is on a flush draw (9 outs, none of his flush outs completes my full house).

The probability of him hitting his flush by the river is Pflush = 1 - (36/45 * 35/44) =~ .36

The probability of me making a full house by the river is Pboat = 1 - (41/45 * 40/44) =~ .17


How do I find the probability of my opponent having the winning hand? I think works out like this but how do you expand this out to find the final probability?

P(makes flush on turn AND I don't make full house on river) OR P(makes flush on river AND I don't make full house on turn)

Thanks.

[jay_shark]

You're interested in the probability he makes the flush given that you do not improve . Lets say you hold 6c-7c and the flop is 7h 6h kd . Your opponent has Ah 2H for the nut flush draw .

Either villain hits his out on the turn followed by a miss from hero , or both miss on the turn and villain hits on the river .

9/45*40/44 + 32/45*9/44 = 32.727%

You also have to factor in the probability that villain hits runner runner for trips or better two pair . The probability of that is just 6c2/45c2 =1.5%

[findingneema]

To win, your opponent must hit his flush and you not fill up. We'll also assume that you won't have a better flush if a fourth suited card hits. So the turn and river must have at least one suited card and one card that doesn't hit your hand.

There are 45c2 = 990 turn+river combos
He makes his flush on 9*8/2 + 9*36 = 360 of them
on 9*4 = 36 of them, you also fill up

So he hits the flush without you filling up on 324 flops, or 32.7%.

[Tom1975]

You can win if either he misses or your both hit.

chances he misses:
36/45*35/44

chances you both hit:
4/45*9/44 + 9/45*4/44

You didn't metion it, but he could also hit running trips to win, the chances of which are:

6/45*2/44

So the final answer is:
36/45*35/44 + 4/45*9/44 + 9/45*4/44 - 6/45*2/44

[jay_shark]

Actually the final answer really depends on the exact hole cards villain has with respect to heros hole cards .

If you look at the example I gave above ,hero has 6c7c and villain has ah-2h and so the exact answer really depends on the cards they have . For instance , if villain had a flush draw like 3h2h , then he won't win if he hits a 3 and a 2 on the turn and river respectively . On the other hand , he could win if he was dealt ah-2h and the turn and river was an ace an a 2 .

Tom , your answer works out to 2/3 which is far off from findingneema and my response .