Advanced Poker Odds Calculations

By now, hopefully you’ve had a chance to read some or all of my numerous posts on calculating hand odds for Texas Hold’em. Let’s have a look at some more advanced calculations, including determining the number of possible specific hands. (Warning: math ahead. But it’s easy, I swear.) With cards, order of dealing sort of matters. As far as betting goes, the order cards appear makes a difference to your betting. Obviously, an A-A in the pocket is more beneficial than A-x followed by an A in the flop. But as far as determining the value of your hand, the order of the 5 cards you get (2 in the pocket, 3 from any of the five community cards) doesn’t matter. So how do we factor this in?

First, we need some notation. In the mathematical field of game theory, two important terms are used: combinations and permutations. They are both from the mathematical field of combinatorics and optimization, which is a subset of statistics, and a superset of game theory.

A combination of objects doesn’t care about the order in which they are picked. A permutation does. For example, the number 967 is made of the digits 9, 6, and 7. So is 796, but it is a different permutation than 967, because order matters.

To calculate the number combinations or permutations related to a poker hand, we first need to understand the factorial notation. Don’t let the math terms scare you. It’s actually a really simple concept:

Factorial notation: n! = n x (n-1) x (n-2) x … x 1
So:
0! = 1 [Don’t worry; it’s just the definition]
1! = 1
2! = 2×1 = 2
3! = 3×2x1 = 6
4! = 4×3x2×1 = 24
etc.

Pretty easy. But the other calculations can get kind of confusing from here. Although, if you understand the basics, you can figure out pretty much any hand. What I’ll do is just cover some starting points for combinations and permutations, then give specific examples in a near future post. Depending on your relationship with math, you may want  to read this article over a couple of times over a few days, to let the concepts sink in. As I learned years ago, it’s easier if you just accept the definitions.

Let’s start with the standard deck. There are 52 cards, of course, and it takes 5 cards to make up a hand. Let’s ignore other players for now. I’ll cover that in a later post, as it involves more complex calculations.

How many possible ways can we form a 5-card hand from 52 cards, where order matters? Well, there are 52 cards to choose from for the first card. Then there are 52-1 = 51 cards for the second card, etc. So the total number of order-specific 5-card hands in a deck of 52 is:

52 x51 x50 x49 x48 = a very big number

But we don’t care about order for now, so we have to remove  duplicates. Before we do that, let’s go over a bit more notation, which will make it easier to follow along.

Permutation notation: P(n,m) = n! / (n-m)!
This is the number of permutations of n objects picked m at a time. That is, order matters. So:

P(52, 5) = number of 5-card hands possible from a deck of 52, where order matters = 52! / (52-5)! = 52! /47! = (52 x51 x50 x49 x48 x47 … x1)/ (47 x46 x45 … x1) = 52 x51 x50 x49 x48 = 311,875,200 hands.

Of course, in terms of the value of a 5-card poker hand, the order does not matter. So we have to employ a more refined formula called the binomial coefficient, to remove duplicate counts:

Binomial coefficient: [n m] = B(n,m) = P(n,m)/m! = n! /m! (n-m)!.

So, the total possible number of unique 5-card hands from a deck of 52 is:

52! / (5! (52-5)!) = 52! / (5! x47!) = (52 x51 x50 x49 x48) /(5×4x3×2x1) = 2,598,960 unique hands. (In case you’re wondering, this is 1/120th of the larger value above.)

That’s it. If you understand the notation above, you can go on to figure out more complex odds, which we’ll discuss in later posts.

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