Poker Math
Most of the math is done away from the tables
By Andy Bloch
Most of the hard core calculations I do away from the table. Then
I take the results and try to form some general rules or simpler
procedures for estimating my odds.
I've written several different programs of my own and use them
and some programs that others have written, (such as http://www.twodimes.net/poker/
) for calculating poker odds and solving simplified poker games.
I've done enough calculations of pre-flop situations that I don't
have to do much calculation. For example, if it's heads up I know
that I have odds to call an all-in raise or reraise with practically
any A if I'm getting pot odds of better than 3:1 -- even if my opponent
will only make that play with QQ, KK, or AA. ("Pot odds"
is the size of the pot divided by the size of the bet that has to
be called. For example, suppose the pot is 1000, my opponent bets
all-in for 500 more. The pot is now 1500, and I have to call 500,
so the pot odds are 3:1. If I have a 25% chance of winning or better,
the correct play is to call.)
Well, to be exact, if the pot odds are exactly 3:1, I should throw
away A2, A3, and A6 through AQ offsuit. But if the pot is just slightly
larger, or if my opponent might also have JJ or AK, then I'm supposed
to call. In fact, getting exactly 3:1 against an opponent whom you
know has either QQ, KK, AA, or AK, you're getting the right pot
odds to call with quite a lot of ugly-looking hands. Any pair. Any
AK. Any suited cards except K2s through K9s, Q2s, and J2s. Offsuit
43 through JT, 53 through T8. And offsuit AT, A5, A4, and A3.
At the table, do I calculate exactly what my probability of winning
is pre-flop? No. What I do know is that if I raise preflop I will
almost never fold to a single all-in reraise when I'm getting 3:1
or even 2:1, if I had something worth stealing with.
On the flop, I know that I'll win half the time if I have 13 outs
("outs" are all the cards that you win with against a
better hand). A good approximation for your chance of winning is
2% per out on the turn and twice that, 4% per out, on the flop.
This will usually underestimate your probability of winning, but
being a little conservative is OK, especially since one or more
of your outs might be in your opponent's hand or might help him
also.
The next step is to think about the hands that your opponent is
likely to hold and assigning some rough probabilities to them. Total
the chance of winning against each of them weighted by their probabilities.
Maybe your opponent is bluffing or semi-bluffing, or maybe you think
you've picked up a tell that makes you think your opponent has a
strong hand -- adjust accordingly. You don't need to do a perfect
calculation, just close enough so you're only making at most a small
mistake. Thinking too hard to get a more exact answer will make
it harder for you to concentrate on other important things like
looking for tells and remembering your opponents' tendencies.
If you're playing against a strong opponent, you might want to
use game theory to determine whether you should call or fold, especially
on the river. If you have a medium-strength hand, game theory says
you should call often enough to make your opponent almost indifferent
to bluffing with clearly losing hands. If your opponent bets the
pot on the river, you should call roughly half the time. If he bets
a fraction X of the pot, you should call roughly 1/(X+1) of the
time. So ask yourself, is my hand in the top X/(X+1) of the hands
I might have in this situation? If it's close, or if you think your
hand is too well-defined by the betting, you can even resort to
flipping a coin.
That's pretty much all the math you need to do while at the table.
It may seem like too much to some, but after a while you should
get used to it. If you can't do it while you're at the poker table,
practice doing it away from the poker table. After a time, most
situations will become second nature.
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