Just How Important is Mathematics?
don't have to be a mathematician to be a good poker player. It
doesn't even help.
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.
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?
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
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:
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.
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.
pot has 7 chips in it and B can stay for 1. B is offered 7-to-l
odds by the pot.
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.
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.
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.
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
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.
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
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.
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.