Multi-Player Advanced Poker Hand Odds – The Magic Formula

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In the past three articles, I’ve covered some fairly advanced calculations for determining hand odds for poker, particularly for Texas Hold’em. In this post, I want to cover the next level of calculations: factoring in multiple players. I’ll start with this post covering the basic mathematical formula, and in the next post I’ll provide an advanced example.


To discuss multi-player poker hand odds, I need to introduce a variation of the binomial formula called the multinomial formula. With this latter formula, you can pretty much calculate the odds of any hand, for any number of players, any number of cards per hand, in basically any type of card game. Provided you understand the formula.


Before you go any further, I’ll warn you that the math here is even heavier than in the past three articles. If you haven’t read them, please do so. (See the refernce list at the end of this post, written in chronological order.) If you decide to skip this post, take one piece of information away with you: All of the previous formulas are simply a guideline for the odds of a hand. In real poker, with multiple players, the previous formulas give you the highest possible odds. The formula presented in this post will give you more accurate odds, which are always small because of the multiple players.


Recall the binomial formula: B(n,m) = P(n,m)/m! = n! /m! (n-m)! = the number of combinations of m objects taken from a set of n objects, and for which order of choosing does not matter. In the case of a standard deck of cards, n = 52.


So if you wanted to know number of all possible 5-card hands in Texas Hold’em, it would be, as we learned previously, B(52,5) = P(52,5) = 52! /5! (52-5)! = 52!/ 5! 47! = (52×51 x49×48 x… x2×1) /(5×4 x3×2 x1) (47×46 x45 x… x2×1) = (52×51 x49×48) /(5×4 x3×2 x1) = 2,598,960 unique hands, where order does not matter. More


The multinomial formula is as follows: M(n, m1, m2, m3, …, mk) = n!/ m1! m2! m3! mk! Since we’re talking about poker, the m values represent the number of cards in each hand dealt. We also need to factor the burn cards typically discarded in Texas Hold’em.






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