Multi-Player Advanced Poker Hand Odds - The Magic Formula

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 x48 x… x2 x1) /(5 x4 x3 x2 x1) (47 x46 x45 x… x2 x1) = (52×51 x49 x48) /(5 x4 x3 x2 x1) = 2,598,960 unique hands, where order does not matter.

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.

Let’s take a specific example, say 5 players for Texas Hold’em, and review the rules so that it’s easierto follow the formula. After the blinds (small and big) are bet, each player is dealt 2 cards, starting to the dealer’s left. Then the next card on the top of the deck is “burned”, followed by the turn of the first three cards of the flop. The next four cards are: burned, turned, burned, turned.

So the set of m-values is {2, 2, 2, 2, 2, 1, 3, 1, 1, 1, 1}, even though there are only 5 players. For each additional player, you have to add another “2″ to the set. Now we can make a relatively simple calculation.

How many many possible ways a deck of 52 cards be dealt out to 5 players, according to the rules of Texas Hold’em, and where the order of cards in a player’s hand does not matter? The order of cards between players also does  not matter, at least not for the initial calculations we’ll consider. We’re treating the separate burn and turn cards as if they are players, too.

The answer is M(52, 2, 2, 2, 2, 2, 1, 3, 1, 1, 1, 1) = 52! /2! 2! 2! 2! 2! 1! 3! 1! 1! 1! 1! = 52! /(2^5) 3! = a very large number, with 66 digits. A lot larger than the number of 5-card hands we saw previously (311 million-plus

You might think that the “1″ values are not doing anything, since 1! = 1, but consider this. If you grouped the burn and flop cards (including 4th and 5th street) into one group, for a total of 1+3+1+1+1+1 = 8, you would eliminate the combinations where a certain card ends up in a turn pile instead of in the community cards. You’d also eliminate combinations where one card shows in the flop rather than in, say, 4th street, or vice versa. We want the extra-group order to be important, because where a card appears in the flop, 4th or 5th streets matters to the betting in Texas Hold’em.

So if we grouped those cards, we’d be eliminating too many combinations that we want to count. Thus the formula as it stands above ensures that we count all unique combinations, but where order of cards within a group doesn’t matter.

I know that’s not much of an example, but it’s a start. I’ll get in to some detailed examples in the next part of this series (which may be two posts ahead). If you can handle the math, probably 4th-year university Calculus or beyond, there is an extremely advanced math paper (PDF) that discusses a poker player’s “risk of ruin”. (It’s partially based on the size of the bank roll you start with, and your win rate.)

Internal references: (chronological order)

(1) The Odds-On Favourite - Calculating Your Chances At Texas Holdem;
(2) The Odds-On Favourite - Part 2;
(3) Can Card-Counting Be Applied To Poker? - The Odds-On Favourite - Part 3;
(4) Do Your Hand Odds Change Depending On The Number of Players?;
(5) Texas Hold’em Poker Hand Odds - A Primer;
(6) Advanced Poker Odds Calculations;
(7) Advanced Poker Odds Calculations - Part 2.

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