What is the probabilty that any of your opponents has any of specifict hands?

Question

I'm creating a database of all choices of 7 cards out of 52. It's quite flexible because it allows to compare choices like:

MyHand+FLOP+TURN+RIVER and OpponentsHand+FLOP+TURN+RIVER

and tell whether my hand is winning or it's a tie. But I'm facing difficulties in trying to identify a probability that any of my four opponents has a hand from a list of my unwanted hands. For example:

I have MyHand = AK, FLOP = JAK, TURN = 9, RIVER = 9. So my list of unwanted hands is:

[10Q, JJ, KK, AA, any hand that has 9s] because in case anyone had it, I would lose. Is there a simple & quick way to count a probability that I lose against any of my four oponents? I ignore flushes for simplicity.

Answer 1

One way that you can accomplish this is to run simulations.

if you would like to do this yourself, you can give your 4 opponents random hands (or select from a range that you would think they would actually play) and keep track of who wins over thousands or millions of iterations with a program. The probability that you lose the hand will end up being the same as the probability that one of your four opponents holds a hand in the range of hands that beats your hand (this would include flushes).

There is a free program called pokerstove that already does this if you are just interested in getting the result. just input the flop, turn, and river as well as your hand and your opponents ranges (or random hand) and it will run millions of simulations and give your probability of winning.

Answer 2

I assume you're talking about Texas Hold 'Em (you probably should put that as a tag and/or mention it in the question, if that is the case).

One tool that might want to look into is the Inclusion-exclusion principle. This does require you keep track of not only the number of ways that a player can beat you, but the number of ways two, three, or four can beat you, so you'll have to distinguish between mutually exclusive outs (for instance, in the example you gave, there aren't enough A out for two players to both get them) and nonexclusive outs (for instance, it's possible for two players to both get 10Q).