Poker And Combinatorics

Probability rocks because it helps us make predictions about the future based on past data. As a pretty good poker player, probability allows me to track patterns to predict a range of hands that my opponents might have.

In No Limit Texas Hold ‘Em, Doyle Brunson wrote a book a while back called the Super System where you make “continuation bets” after the flop if you’re the pre-flop raiser. Using probability, I track how many “wet boards” (flops that can make flushes and straights) and “dry boards” (flops like 2, 7, K where two pairs and sets and pre-flop overpairs would be valuable). Does my opponent know when to make “continuation bets” on wet boards versus dry boards? How does their position relative to the dealer affect their play? I memorize all this data to make predictions on this opponents range of hands.

There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million). How do we know this? Do we come us with this math from permutations or combinations?

But first let’s go over the basic terms in probability. Permutations are the number of different possible ways we can arrange a set of elements. A factorial is represented by the sign (!). When we encounter n! (known as ‘n factorial’) we say that a factorial is the product of all the whole numbers between and n, where n must always be positive.

n! = n × (n - 1) × ......3 × 2 × 1

Below is the mathematical equation for a permutation where a ‘r' count of the different arrangement which can be made from the given set of ‘n' things.

Next is the combination mathematical formula: the combination of ‘r‘ things from the available 'n' things would be factorial of n, divided by the product of the factorial of r and factorial of (n - r).

So when do we use permutations versus combination in probability? We use permutations when order/sequence of arrangement is needed and when we’re looking at different things. The permutation of 2 things from 3 given things a, b, c is ab, ba, bc, cb, ac, ca. On the other hand, we use combination to find the number of possible groups which can be formed and when we’re looking at similar things. The combination of 2 things from 3 given things a, b, c is ab, bc, ca.

Now that we ‘kinda sorta’ have some grasp of basic probability, we can go back and answer the question above as to whether poker is more about permutations or combinations. Poker math derives from combinations. First of all, the order that the cards are dealt in does not matter. Second of all, there are 52 cards in a poker deck, and a hand is a combination of 5 of those cards. We don’t need permutation because we want to eliminate duplicates. For example, ace of diamonds and ace of spades don’t need to be counted twice; we’re only interested that the two aces go with three Kings to form a full house. Therefore, the number of possible poker hands is n equals 52 for 52 cards in the deck and r equals 5 for five cards in your hand.

We plug n and r in and 52! / (52 – 5)! * 5! = 52! / 47! * 5! = 2.59 million possible poker hands.

In my next post, I’ll take this a step further and go through the map to finding winning hands. In a later post, I’ll go into binomial coefficients  to calculate specific combination of cards.