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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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