The Combinatorial Function

By Bud Kelly
November 2016

The combinatorial function is the foundation of all poker probability calculations. It is not difficult to learn, and with a little practice you can have it on hand as a valuable tool in your toolbox.

In plain language, the combinatorial function tells you how many ways you can arrange m objects, taken n at a time. (The order of the objects does not count.) For example, how many ways can you arrange 52 cards, taken 2 at a time, like the down cards in Texas Hold'em. Here is the math:

C(n, m) = n! / m! (n-m)!

If you are not mathematically inclined, don't let the mathematical notation intimidate you. In fact, it turns out to be quite simple in practice. The only thing that might be a bit unfamiliar to you is the "!" symbol, which is used in mathematics to signify the factorial function.

The factorial function means that you multiply the number by every number preceding it, all the way down to 1.

For example, 3! would be 1 x 2 x 3, or 6.

As another example, 6! would be 1 x 2 x 3 x 4 x 5 x 6, or 720.

You can see that any reasonably large number will become huge when the factorial function is applied, but let not your heart be troubled. As it turns out, using the combinatorial function generally allows you to cancel out a lot of the multiplication and make the calculation simple enough to do in your head. That is, with a little practice.

Let's take the example of how many ways you can arrange 52 cards, taken 2 at a time, like the down cards in Texas Hold'em. In effect, how many different down card hands are possible. Here n = 52, and m = 2. So the combinatorial function calculation is:

C (52, 2) = 52! / 2! (52-2)!

C (52, 2) = 52! / 2! (50)!

C (52, 2) = 1 x 2 x 3 …..x 49 x 50 x 51 x 52 / 1 x 2 (1 x 2 x 3… x 49x 50)

Now you can simplify this by cancelling out numbers common to the numerator and denominator:

C(52, 2) = 51, x 50 / 1 x 2

C(52, 2) = 1,326

Even if you are averse to doing the math, just remember the number 1,326, because it is the number of possible 2 card hands you can be dealt in Texas Hold'em.

1,326 Possible Starting Hands

Since card suits all have the same rank in poker, many of these hands are equivalent to each other. (For example, any pair of queens has the same rank as any other pair of queens, no matter what suits they are.) As it turns out, though there are 1,326 possible starting hands, there are only 169 non-equivalent hands. That makes things a little simpler.

Of these 169 equivalent hands, 13 are pairs.

Of these 169 equivalent hands, 78 are non- suited.

Of these 169 equivalent hands, 78 are suited.

Note that 13 + 78 + 78 = 169.

13 Pairs, 78 Non-Suited, 78 Suited.

Just memorizing this will give you a good grasp of what the prospects are for any starting hand you are dealt. But a better grasp can be gained by memorizing the probability of being dealt various hands:

Probability of a pair, any pair, 1 in 17.

Probability of 2 suited cards, any suit, 4 in 17.

Probability of 2 unsuited cards, any suit, 12 in 17.

Note that 1/17 + 4/17 + 12/17 = 17/17, representing all of the possibilities.

Pairs 1/17, suited 4/17, unsuited 12/17.

You can see that being dealt a pair is fairly fortunate. So is being dealt two suited cards. Between these two possibilities, and you are enjoying one of the top 5 out of 17 hands. Prospects will be good, but it all depends on the specific cards, the number of players, and of course the cards that they have been dealt.

So, unfortunately it's not that simple. The actual cards you have been dealt, the number of players, and the cards that they have been dealt create enormous complications in evaluating the prospects of winning with your hand.

Fortunately, a number of poker gurus, theorists, and people who have spent their lives playing Texas Hold'em, have published systems of evaluation. These systems differ slightly from author to author, but knowing something about most of them, and everything about any one of them, will put you in a good position to evaluate your prospects and guide your decisions in the opening round of betting.

But of course, at the highest levels, Texas Hold'em is not about playing cards but about playing people. It's "a game of people, played with cards." That's where the greatest challenge lies, as well as the greatest potential rewards.

 

Note: Bud Kelly is a writer of fiction, non-fiction, magazine articles, news columns, poetry, songs, music, and jokes. You can see some of his work on Nook and Kindle. If there's anything you would like written for your website, contact us via the link on the menu and we will put you in touch with Bud.

 

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