Using Mathematics to Read Hands

When you can't actually put a
person on a hand but have reduced his possible hands to a limited number, you
try to use mathematics to determine the chances of his having certain hands
rather than others. Then you decide what kind of hand you must have to continue
playing. Using mathematics is particularly important in draw poker, where your
main clue to what an opponent might have is what you know about his opening,
calling, and raising requirements.

If, for example, you know an opponent will raise you with three 2s or better
before the draw, you can resort to mathematics to determine what hand is
favored to have him beat. It works out to something like three queens.
Obviously, then, if you have three 3s, it's not worth calling that opponent's
raise on the chance that he has specifically three 2s. But if you have
something like three Ss or three 6s, the pot odds make it correct to call
because now not only might you draw out on a better hand by making a full house
or four-of-a-kind, but there are a few hands your opponent could have which you
already have beat.

Sometimes you can use a mathematical procedure based on Bayes' Theorem to
determine the chances an opponent has one or another hand. After deciding upon
the kinds of hands your opponent would be betting in a particular situation,
you determine the probability of his holding each of those hands. Then you
compare those probabilities. If, for instance, in draw poker you know a particular
player will open either with three-of-a-kind or two pair but will not open with
one pair and will check as a slow play with a pat hand, then it is 5-to-2
against that player's having trips when he does open. Why is this so? On
average, according to draw poker distribution, a player will be dealt two pair
5 percent of the time and trips 2 percent of the time. When you compare these
two percentages, you arrive at a ratio of 5-to-2. Therefore, the player is a
5-to-2 favorite to have two pair.

Let's say in hold 'em an opponent puts in a big raise before the flop, and you
read him for the type of player who will raise only with two aces, two kings,
or ace, king. The probability that a player gets two aces on the first two
cards is 0.45 percent. The probability of his getting two kings is also 0.45
percent. So he will get two aces or two kings 0.9 percent of the time on
average. The probability of his getting an ace, king is 1.2 percent. By
comparing these two probabilities -1.2 percent and.9 percent you deduce that
the chances are 4-to-3 in favor of your opponent's having ace, king rather than
two aces or two kings. Of course, knowing your opponent is a 4-to-3 favorite to
have ace, king is not enough by itself to justify calling his raise with, say,
two queens. You are a small favorite if he does have ace, king, but you're a
big underdog if he has two aces or two kings. Nevertheless, the more you know
about the chances of an opponent's having one hand rather than another when he
bets or raises, the easier it is for you to decide whether to fold, call, or
raise.

Earlier in this page we talked about a player in seven-card stud raising on third street with a
king showing, and we pointed out that he might have two kings, but he might
also have a small pair or a three-flush or something like J,Q,K. To simplify,
we'll assume you know this particular player will raise only with a pair of
kings or a three-flush. You have a pair of queens. The probability is about 11
percent before the raise that your opponent has another king in the hole to
make a pair of kings, and it's about 5 percent that he has three of the same
suit. This is simply the mathematical probability based on card distribution
and has nothing to do with any action the player takes. Therefore, when your opponent
raises, which now limits his possible hands on the basis of what you know about
him to either two kings or a three flush, he is an 11-to-5 favorite to have the
two kings, and you would probably fold your two queens. However, another king
showing somewhere on the table radically reduces the mathematical probability
of your opponent's having two kings before he raises because there are only two
kings instead of three among the unseen cards. The probability of your
opponent's having two kings is cut to about 71/z percent. A raise now makes it
about 40 percent that your opponent has a three-flush rather than two kings.
Depending upon your position, your queens may be strong enough to justify a
call. In this case you read your opponent's hand not just on the basis of what
you know about him, the action he takes, and the exposed card you see, but also
on the basis of a mathematical comparison of his possible hands.

It does not, of course, take a mathematical genius to realize that another king
on the table decreases the chances of an opponent's having two kings before he
raises, so using math to read hands does not always require the precise
knowledge of card-distribution probabilities presented here. Furthermore, you
need to complement mathematical conclusions with what you know about a player.
For example, in a relatively small-ante games, some players might not raise
with two kings when there is no other king showing in hopes of making a big
hand, but they will raise with two kings when there is a king showing to try to
win the pot right there. They decide to go for the pot right away precisely
because of the presence of that other king, which reduces their chances of
improving. When you are up against such players, the presence of another king
might actually increase the probability of their having two kings after they
raise - not on the basis of mathematics but on the basis of the action they
have taken and what you know about the way they play.

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