Poker Math Made Simple

In my writings I try to take a math-based approach to the game without actually going over too much – well – actual math. For the type of games I write for, getting the math exactly right to the decimal place is often unnecessary. It’s more important to be able to think about the game in a logically consistent manner, as once you can do that consistently, you will find most of the correct plays. Playing with numbers will just give you an idea of exactly how correct your play was.

Before I go further, I want to emphasize that this approach works fine in small-stakes cash games, but in high-stakes games, in online games, and in tournaments, it’s insufficient. In those venues, you’ve got to get dirty with the actual numbers if you want to succeed.

But there is one sort of math that I think is useful (and fun) in small-stakes games—combination math. This math has the virtue of being relatively simple to understand. Also it can help you answer all those, “How often does that happen?” questions that we all have sometimes after a funny hand.

The Basics

There are 1,326 possible starting hands. How do you get that number? It’s simple with a special mathematical operation called a combination.

You want to know how many combinations of two cards you can make from a 52-card pack. That is, you are trying to choose two cards from 52 in all possible ways. Just like you can say X plus Y or X times Y, you can say X choose Y. In this case, the number we want is 52 choose two.

For the lazy, you can actually type “52 choose two” directly into Google, and it will give you the answer. If you want to know how to calculate the “choose” operation, it’s a little complicated. You take the first number, count it down by one as many times as the second number, and multiply them together. So for 52 choose two, it’s 52 × 51. If instead we wanted 52 choose four, it would be 52 × 51 × 50 × 49.

That’s the numerator. The denominator is the second number counted down by one all the way to 1 and the multiplied together. So for 52 choose two it’s 2 × 1, and for 52 choose four it’s 4 × 3 × 2 × 1.

To get the final answer, you take the first product and divide by the second product. So 52 choose two is (52 × 51) / (2 × 1), which is equal to 1,326. The more complicated 52 choose four is (52 × 51 × 50 × 49) / (4 × 3 × 2 × 1) which equals 270,725. You can get that number by multiplying and dividing, or by typing “52 choose four” into Google.

How Many Flops?

So here’s a question. How many possible hold’em flops are there? Well, there are 52 cards in the deck, and we can choose any three for the flop. So the answer is 52 choose three. Google tells me that number is 22,100.

You may have seen a different number, 19,600, quoted elsewhere for possible flops. That number is also correct—but it’s the answer to a slightly different question than the one I asked. That question is, “Given that I have my hand, how many possible flops are there?”

The “given my hand” part is the twist, and it’s an important one. Once you have a hand, you know two of the cards. And the deck has only 50 cards remaining. So now instead of choosing three cards from 52, we are choosing three cards from only 50. If you type “50 choose three” into Google, you get 19,600.

If I Have A Suited Hand, How Often Do I Flop A Flush?

So here’s a question most hold’em players want to know the answer to. Figuring it out is a little more complicated than the math we’ve done so far.

The probability of flopping a flush is equal to the total number of flops that give us a flush, divided by the total number of flops (given our two cards). We’ve already figured out that the total number of flops given our hand is 19,600. So we just have to figure out the first part—how many flops give us flushes.

If we have two diamonds in our hand, then there are 11 left in the deck. To flop a flush, we need three of those diamonds on the board. So the answer is 11 choose three. I type “11 choose three” into Google, and I get 165. That’s how many flush flops there are.

Just divide that by 19,600, and you have your chance, which is about 0.84%.

If I Have A Suited Hand, How Often Do I Flop A Flush Draw?

I’ve upped the difficulty once more. The process is the same, except that now I need two diamonds on the flop, and one non-diamond. Here’s how you count how many of those flops exist.

First we need two diamonds. There are 11 left in the deck, so we have 11 choose two or 55 combinations of those. Now we need a third card that is not a diamond. There are 13 diamonds in the deck, so there are 52 minus 13 or 39 non-diamonds. To count as a flush draw, any old non-diamond will do. So we just multiply our 55 flush draw combinations by the 39 possible side cards. That’s 2,145 possible flops.

Now we divide that by 19,600 and we get a 10.9 percent chance to flop a flush draw with two suited cards.

All Red Or All Black?

There’s a prop bet out there about whether the flop will be all one color or will be mixed colors. What’s the chance a flop comes all one color?

The easiest way to figure this out is a little different than what we’ve done before. First, pick a card for the flop. It can be any card, red or black, it doesn’t matter. Let’s say it’s red. There are 51 choose two possible combinations of second and third cards. That’s 1,275 possible flops once we’ve dealt out the first card.

Of those, we want to know what percentage are all red. There are 25 red cards left, so it’s 25 choose two or 300. Therefore, given that the first card is red, the chance of an all-red flop is 300 / 1275 or 23.5 percent.

Similarly, if the first card is black, the chance of an all-black flop is also 23.5 percent. This means that no matter what card comes first, the chance is the same. Therefore, 23.5 percent is also the overall chance that a flop will be all one color. You can offer an opponent 3:1 on all one color and have a nice edge.

Final Thoughts

This simple mathematical operation, “choose,” can help you work out most questions about how often things happen at the poker table. Try it yourself. It’s fun. ♠

Ed’s newest book, The Course: Serious Hold ‘Em Strategy For Smart Players is available now at his website edmillerpoker.com. You can also find original articles and instructional videos by Ed at the training site redchippoker.com.