How does software work that calculates the winning probability of Texas Hold'em or Omaha hands versus 8 random opponent hands?

Question

So, there are Texas Hold'em computer games where you play up to 8 opponents, and presumably some of these computer games tell you that your probability of winning is provided that your opponents are random. If someone does not know, in Hold'em, each player is dealt 2 private cards, and then on average 5 community cards are dealt in the middle (the first 3, then 1, then another 1), and the winner is the player who can make the best 5 - a card poker combination in which they can use any combination of 2 personal cards and 5 community cards. In Omaha, each player is dealt 4 private cards, and another 5 community cards, and the winner is the player who can make the best 5-card poker combination using 2 private cards and 3 community cards.

Thus, in Hold'em for any player with a personal hand, there are more than 10 ^ 24 ways that you could manage the private hands of opponents and 5 community cards. So, how do they calculate / evaluate your probability of winning at the beginning if your opponents 8 hands are random? The situation in Omaha is even worse, although I have never seen a computer game in Omaha that actually gives you your chances against the hands of 8 random opponents. But anyway, are there any software tricks that these probability calculations of winning can get (or say, fix within 3 or 4 decimal places) faster than brute force? I hope someone can answer here, who wrote such a program before, quickly enough, so I ask here. And I hope that the answer does not imply an estimate of random samples, because there is always a small chance, which may be aloof.

Answer 1

As you have determined, the expected winning rate is a complex summation summation and should be approximated. The standard approach is to use the Monte Carlo method, which includes the simultaneous modeling of different hands and the adoption of an empirical average value: # wins / # games.

Interestingly, the error (MSE) of this approximation does not depend on the dimension (number of combinations), in particular, if X = 1, if you win, 0 if you lose, MSE = var (X) / N = p * (1- p) / N, where p = Prob (X = 1) (unknown), and N is the number of samples.

There are a number of different Monte Carlo methods that can improve the variance of the vanilla sampling approach, such as importance sampling, regular random numbers, Rao-Blackwellization, control variations, and stratified sampling, to name just a few.

edit: just saw that you are looking for an approach based on a nonrandom approximation, I doubt that you will have much luck in approaches with determinate approximations. I know that the current state of computing in poker research uses Monte Carlo methods to compute these probabilities, albeit with a few tricks to reduce variance.

Regarding "because there is always a small chance that can be removed", you can always get the probability of a high probability of an error with Hoffding's inequality.

Answer 2

I would use a pre-computed coefficient table instead of on-the-fly calculation. The tables that list them are extremely easy to find and have been around for quite some time, so they are proven tools. It would be quite simple to compare your open cards + community cards with the percentage indicated in the pre-calculated table, and immediately return the value to you, skipping the calculation time on the fly.

There are only 52 cards in the deck (classically), therefore, if you just find all the possible solutions in advance, it’s much faster to read them and not to re-calculate the odds for each hand on the fly.

Here is a link to an incomplete coefficient table: http://www.learn-texas-holdem.com/texas-holdem-odds-probabilities.htm

I would think of it as a password crack. Instead of rudely forcing each character individually, use a shared password list to reduce the calculation time. The difference in this case is that you know every possible combination ahead of time.