Poker Math - Part I

Everyone is afraid of math. Teenagers hate algebra, checkout clerks shut down without a calculator, politicians hire teams of "experts" to crunch their numbers, and poker players don't want to hear about odds and equity – or at least a lot of them don't. The very idea that mathematics is useful in poker brings out righteous indignation from a substantial number of today's playing population.

I'm not afraid of math, and I'm going to use the pages of this magazine to show that nobody else should be afraid of math, either. I was a mathematics major at Yale University, but I definitely don't use complex analysis, differential equations, or algebraic topology at the poker table. Trust me, most of the math that's used in poker can be understood by anyone with a high school education. There are people out there pursuing advanced game theory studies about poker, but we won't get into those. My purpose is to teach the average player everything he needs to know to be wholly math-literate at the table.

Let's start by looking at some of the most basic terms used in poker math:

1. Odds

Some gamblers might have heard this word over and over again without really knowing what it meant. Odds are a cousin of probability. So, what's probability? Probability is the chance that a given event will take place. When the weatherman says there is a 25 percent chance of rain today, he is expressing a probability. He is saying the probability that it will rain today is 25 percent. What that means is that if today happened 100 times, 25 of those times it would rain, and 75 times it wouldn't. This brings us back to odds. Odds compare the number of times an event will happen to the number of times it won't. In our weatherman example, the odds against rain falling today would be 75 to 25 – that is, for every 75 times that it would rain, it wouldn't rain 25 times. We write these odds, 75-25. It is equivalent to express these odds as 3-1, because we can see that for every time it rains, it doesn't rain three times (75 divided by 25 equals 3).

Let's look at the probabilities and the odds for some different events.

Coin flip, heads: probability 50 percent, odds 1-1

Airline flight delayed: probability 12.5 percent (data from Bureau of Transportation Statistics), Odds 7-1

Picking the A out of a deck: Probability 1/52 = 1.9 percent, Odds 51-1 (in this case, it's easier to do the odds)

2. Combinations

In a game like Texas hold'em, we are interested in questions like, "what are the odds against completing a four-card flush draw after the flop?" This is a much harder question than, "What are the odds against completing a four-card flush draw after the turn?" In the latter case, there is only one card left to come. There are 46 unknown cards at that point (52 minus the two in our hand and the four on the board). So, to calculate our odds of making a flush draw after the turn, we just compare the number of unknown cards that don't help us (37) to the number of unknown cards that do (nine). The odds of making a flush draw after the turn, therefore, are 37-9, or about 4.1-1.

After the flop, with two cards still to come, it's not as straightforward. If we don't make our flush on the turn, we could still make it on the river. How do we account for this? We do it by counting the different combinations of cards that could come. Say we hold the 9 8 and the flop is 10 4 2. The turn and river could be A K. They could be A A. They could be 3 3. They could be J J. Each of these is a different combination of turn and river cards. Note that J J is the same combination as J J, because they result in the same board. Now, instead of counting cards to determine our odds, we count combinations. If you write down every last possible combination for the turn and river in this hand, it turns out that there are 1,081. Then, if you look closely at all of them, it turns out that 378 result in a flush for our hand. So, the odds against making a four-card flush draw after the flop are 703-378 (because 1,081 minus 378 is 703), or about 1.86-1.

Just by learning these two terms, you now know how to calculate the odds against making any hold'em hand after the flop, or after the turn. Cool, huh? It is cool, but it's also a lot of work to calculate your odds for every draw you might run into. Luckily, you don't have to, as I'll explain.

3. Outs

Your outs are the number of cards in the deck that will improve your hand. The flush draw we held above had nine outs. An open-end straight draw has eight outs. Two overcards have six outs. You could go through the odds calculation for each of these draws – or you could just read the results off the chart.

Notice that with 14 outs or more, we're actually more likely than not to improve.

Again, it's not important to know these exact numbers. In fact, there's a useful trick called the Rule of Four to help you. Multiply your number of outs by four, and that number is roughly your percent chance of improving after the flop. So, with a flush draw on the flop, you have about a 9 times 4 = 36 percent chance of improving to a flush by the river (the actual number is 35 percent). Notice that this is the probability of improving, and not the odds against improving. Here are some quick conversions:

25 percent = 3-1 against

33 percent = 2-1 against

40 percent = 3-2 against

50 percent = 1-1 against

If you understand what I've written, you understand everything you need to assess your chances of improving in a hold'em hand. With enough practice, the numbers will become natural enough to you so that you can focus on other things at the table.

Maybe you're still one of those people who think math isn't really useful in poker, that once you know the basic odds, the rest of the skills you need have nothing to do with math. Stay tuned for my next column, as I hope I can change your mind.

Matt Matros finished third in the 2004 World Poker Tour Championship, and cashed four other times in major tournaments last year. His book, The Making of a Poker Player, is on the shelves now.