So there are Texas Hold'em computer games where you play up to 8 opponents and supposedly some of these computer games tell you your probability of winning assuming your opponents hands are all random. In case someone doesn't know, in Hold'em each player is dealt 2 private cards and then eventually 5 community cards are dealt in the middle (first 3, then 1, then 1 more), and the winner is the player who can make the best 5 card poker hand they can using any combination of their 2 private cards and the 5 community cards. In Omaha, each player is dealt 4 private cards and there are still 5 community cards and the winner is the player who can make the best 5 card poker hand using 2 private cards and 3 community cards.

So, in Hold'em, for any given player's private hand, there are over 10^24 ways that 8 opponents' private hands and the 5 community cards could be dealt. So how do they calculate/estimate your probability of you winning in the beginning, assuming your 8 opponents' hands are random? In Omaha the situation is even worse although I've never seen an Omaha computer game that actually gives you your odds against 8 random opponents' hands. But anyway, are there any programming tricks that can get these winning probability calculations done (or say, correct within 3 or 4 decimal places), faster than brute force? I'm hoping someone can answer here who's written such a program before that runs fast enough, hence why I'm asking here. And I'm hoping the answer doesn't involve random sampling estimation, because there's always a small chance that could be way off.
Mar 23, 2022 411 views

The projected win rate, as you noted, is an impossibly high sum that must be approximated. The Monte Carlo method, which involves repeatedly simulating various hands and calculating the empirical average: #wins/#games, is the conventional strategy.
Interestingly, the (MSE) error of this approximation strategy is independent of the dimensionality (number of combinations) specifically, MSE = var(X)/N = p*(1-p)/N where p = Prob(X=1) (unknown), and N is the number of samples.

Importance sampling, common random numbers, Rao-Blackwellization, control variates, and stratified sampling are just a few of the Monte Carlo approaches that can help enhance the variance of the vanilla sampling strategy.

I noticed you're looking for a non-random approximation approach; I doubt you'll have much luck with deterministic approximation approaches; I'm aware that the current state of the art in compute poker research employs Monte Carlo methods to compute these probabilities, albeit with several variance-reduction techniques.
With Hoeffding's inequality, you can always establish a high probability bound on the error rate, even if "there's always a small chance that may be way off."
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