Mathematics and poker

All people are afraid of mathematics. Teenagers hate algebra, cashiers do not freeze when a calculator, politicians hire a team of "experts" to handle their figures, and poker players do not want to hear about probabilities and equity - or at least several of them will not want to hear. Just the idea that mathematics is useful in poker brings outrage from several players.

I'm not afraid of mathematics, and I will use these pages to show that no one should be. I majored in mathematics at Yale University, but I will definitely not use complex analysis, differential equations or algebraic topology poker. Believe me, most of the mathematics used in poker can be understood by all people with a high school diploma. There are people who engage in theories advanced poker game, but we will not go into them. My goal is to teach the average player everything he needs to know to be fully learned mathematics at the table. Start by looking at some basic terms used in the mathematics of poker:

1. Ratings.
Some punters had heard these words again and again, without really knowing what that means. The odds are the cousin of probabilities. So what are the odds? The odds are the odds that an event occurs. When a meteorologist said there was a 25% chance of rain today, it provides a probability. What he says is that the probability of having rain today is 25%. What this means is that if we repeat today 100 times the day it rains 25 times, and it does not rain 75 times. This brings us to dimensions. Ratings compare the number of times an event occurs to the number of times it will not happen. In our example weather, rain against the odds would be 75 to 25 - 75 for each time he will not wet, it will wet 25 times. We write these odds this way: 75-25. This is equivalent to say that the odds are 3-1, because for every time it rains, it will not rain three times (75 divided by 25 = 3).

Look at the probabilities and odds with other examples:

Have battery in battery-or-face: 50% probability; Odds 1-1.
Having your flight delayed: probability of 12.50% (data from the Bureau of Transportation Statistics); odds of 7-1.
Peg the ace of spades in a deck of cards: Probability of 1/52 = 1.9%; odds of 51-1 (in this case, it is easier to calculate the odds that the probability).

2. Combinations
In a game like Texas Hold'Em, we are interested in questions like: "What are the odds to complete his flush draw after the flop? ". This is a much more complicated question: "What are the odds to complete his flush draw after the turn? ". In the latter case, there is only one card to come. There are 46 unknown cards at this time (52 minus the two in your hand and four community cards). So to calculate the odds against the fact complete his flush draw after the turn, we only need to compare the number of unknown cards that we do not help us (37) to the number of unknown cards that help us (9). The odds against the fact complete his flush draw with one card to come are of 37-9, or about 4.1-1.

After the flop, with two cards to come, this is not as simple. If we do not complete our flush draw on the turn, we can complete the river. How would like us all that into account? We do this by calculating the different combinations of cards that can come out. Say we have 9h8h and the flop is Th-4h-2c. The turn and river can be Ah-Ks. They can be Ah-As. They can be 3h-3s. They can be Jc-Jd. Note that Jc-Jd is the same combination that Jd-Jc, since the result among the community cards would be the same. Now, instead of counting cards that determine our ratings, we expect combinations. If all possible combinations on the turn and the river is noted for that hand, one would arrive in 1081. Then, if we look more closely at these combinations in 1081, we realize that 378 of them complete our draw. So the odds against the fact complete our draw is therefore 703-378 (1081 minus 378 = 703), or about 1.86-1.

Just learning these terms, you now know how to calculate the odds for any Hold'Em hand after the flop or after the turn. Great, is not it? Yes, that's great, but it's also a lot of work to calculate the odds of all possible runs. Fortunately, you do not have to do it and I'll explain why.

3. The "outs"
The "outs" are the number of cards in the deck that improve your hand. The flush draw which we spoke earlier, nine "outs." Bilaterally to the sequence (open-ended straight) draw has eight outs. Two overcards (above the flop cards) has six outs. You can calculate the odds of each of these prints ... or just read the results in the following table:

Note that with 14 outs or more, you are actually more likely to improve your hand than otherwise.

It is not important to know the exact figures. In fact, there is a very useful little trick to help you: Rule of four. Multiply your number of outs by 4, and this number will be very close to the percentage chance of improving your hand after the flop. So with a flush draw on the flop, you have about 9 * 4 = 36% chance of complete color by the river. Note that this result gives you the chance of improving his hand, not the fact the odds against improving your hand. Here are some conversions:

25% = 3 against 1.
33% = 2 against 1.
40% = 3 against 2.
50% = 1 against 1.

If you understand what I just wrote, you'll understand everything you need to estimate your chances of improving your hand in hold'em. With enough practice, these figures become so natural that you can concentrate on other things to the table.

Maybe you're still one of those people who believe that mathematics is not really useful in poker and once you know the basic dimensions, the rest of the skills that you need have nothing to do with mathematics. Stay tuned for my next column, I hope to make you change your mind.

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• Mathematics and poker - part 2

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