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Just How Important is Mathematics?


You don't have to be a mathematician to be a good poker player. It doesn't even help.

True, poker offers some of the most fascinating of mathematical problems and for that reason has engaged the attention of the best mathematicians. Some of their researches invade the highest levels of the higher mathematics. Their findings are published in books. You can trust these books. I have read dozens of poker books and as far as I know Oswald Jacoby's is the only one written by a master mathematician, yet I have never seen a poker book in which the quoted odds are wrong by more than some insignificant fraction or percentage. But you do have to have a knowledge of simple arithmetic, a memory for the simple odds that you read about in books, an undertsanding of what these odds mean, and a quick eye for appraising the size of the pot. It is considered neither cricket nor poker to stop and count the pot every time your turn comes and you have to make a decision.

When you have the best hand around the table, and you know or feel sure that you have the best hand, mathematics doesn't enter into it at all. You simply shove your money into the pot. You may take some comfort from the figures, elaborately prepared by mathematicians, proving that the best hand going in is usually the best hand coming out; but what would it matter? Who ever heard of dropping the best hand?

So the only mathematical questions arise when you may not have the best hand going in. In any such case, you must improve to win. You must then ask yourself three questions: 1, What are the odds against my improving? 2, What are the odds offered me by the pot? 3, What is the chance that I will win if I do improve?

The first question is answered by tables of odds that you can quickly and easily commit to memory; nearly every case that may confront you is treated in the closing pages of this website. The second question—the odds offered by the pot—is a matter of an eyecheck of the pot or knowledge of how much is already in it and how much you have to put in. The third question— your chance of winning if you do improve—is answered partly by the tables of probabilities and partly by your knowledge of the game. Here are some examples to clarify the latter:

First Example

Draw poker, seven players, blind opening. Dealer (G) antes 1, A at his left opens blind for 1, B raises blind for 2. C bets 3. D, E, F, G, A drop. B can stay for 1.

B holds 10-9-8-7-K. The odds are 39 to 8 (5 to 1) against his making a straight. The odds are only 27 to 20 (almost even) against his making a pair of sevens or better.

The pot has 7 chips in it and B can stay for 1. B is offered 7-to-l odds by the pot.

If B fills his straight he has at least a 90 percent chance of winning—that is, not once in ten times will C have or draw a hand better than a ten-high straight. On this basis only, B should play; because the odds are only 5 to 1 against making the straight and the pot offers him 7 to 1.

B's chance if he makes a pair depends entirely on what kind of player C is. In a good game B would discount entirely the chance of winning on a low pair, because C would not have bet with less than aces or at least kings. In a liberal game, C might have bet with a four-flush or bobtail straight. In this case a low pair might win; but about 15 percent of the time B might lose even if he makes his straight. Here we must assume a tight game, however, because in a liberal game all the other players would not have dropped.

Second Example

Draw poker, seven players, jacks to open, pass and back in. The ante is 7 chips, the limit 2 before the draw, 4 after the draw. Dealer is G. A, B, C, D pass. E opens for 2 and all players from F through C drop.
D holds 10-9-8-7-K. The odds are still 5 to 1 against his making a straight. The pot offers 9 chips against the 2 he must pay to call, or to 1. He cannot win by pairing because E has at least jacks. The odds against him are greater than the odds he is offered and he throws in his hand.

These are the simplest possible examples (though both of them happen frequently) and in most cases closer figuring will be necessary. The examples were purposely made simple to illustrate the basic theory of the application of mathematics to poker. Mathematics in poker can be very useful—in fact, some knowledge of the odds is essential—but nothing can be more damaging than placing slavish reliance in the mathematical probabilities. Events always alter the a priori assumptions. For example, in a seven-hand game of draw poker it is useful to know that two aces should be the best hand, normally, before the draw; but if you hold the aces and three players have already come in before you, you must assume or at least suspect that your two aces are not the best hand; and if one of those players has raised, you can be fairly sure that they are not the best hand.

The mathematical expectancies must also be modified by a further question you must ask yourself: "Is there any point to betting?" For example, you are against one other player in a draw poker game. He draws three cards, and you draw three cards to two jacks. You make jacks up. The odds are 21/2 to 1 that he did not improve his pair, so mathematically you have a good bet. But the realities are that if he did not improve he probably will not call and your bet becomes pointless, and if he did improve and calls he can probably beat you. Therefore, mathematics or no mathematics, you do not bet. If you had made three jacks you would have bet, because mathematics tells you that the odds are 8 to 1 against his having made three of a kind and you may get a call if he made aces or kings up.

When you are deciding whether to stay or drop, and when betting is normal, mathematics is an excellent guide. When players begin raising and reraising, mathematics goes out the window.

Nevertheless every accomplished poker player should know the odds against improving on various draws and should not forget to compare those odds against the odds being offered by the pot. This may seem so fundamental that it is hardly worth mentioning, but not one poker player in a hundred bothers to do it and the vast majority of all losses suffered in poker games can be attributed to sticking around when the pot offers shorter odds than the odds against improvement.

 
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