MATHEMATICAL EXPECTATION IN POKER
Poker plays can also be analyzed in terms of expectation.
You may think that a particular play is profitable, but sometimes
it may not be the best play because an alternative play is
more profitable. Let's say you have a full house in five-card
draw. A player ahead of you bets. You know that if you raise,
that player will call. So raising appears to be the best play.
However, when you raise, the two players behind you will surely
fold. On the other hand, if you call the first bettor, you
feel fairly confident that the two players behind you will
also call. By raising, you gain one unit, but by only calling
you gain two. Therefore, calling has the higher positive expectation
and is the better play.
Here is a similar but slightly more complicated situation.
On the last card in a seven-card stud hand, you make a flush.
The player ahead of you, whom you read to have two pair, bets,
and there is a player behind you still in the hand, whom you
know you have beat. If you raise, the player behind you will
fold. Furthermore, the initial bettor will probably also fold
if he in fact does have only two pair; but if he made a full
house, he will reraise. In this instance, then, raising not
only gives you no positive expectation, but it's actually
a play with negative expectation. For if the initial bettor
has a full house and reraise, the play costs you two units
if you call his reraise and one unit if you fold.
Taking this example a step further: If you do not make the
flush on the last card and the player ahead of you bets, you
might raise against certain opponents! Following the logic
of the situation when you did make the flush, the player behind
you will fold, and if the initial bettor has only two pair,
he too may fold. Whether the play has positive expectation
(or less negative expectation than folding) depends upon the
odds you are getting for your money - that is, the size of
the pot - and your estimate of the chances that the initial
bettor does not have a full house and will throw away two
pair. Making the latter estimate requires, of course, the
ability to read hands and to read players, which I discuss
in later pages. At this level, expectation becomes much more
complicated than it was when you were just flipping a coin.
Mathematical expectation can also show that one poker play
is less unprofitable than another. If, for instance, you think
you will average losing 75 cents, including the ante, by playing
a hand, you should play on because that is better than folding
if the ante is a dollar.
Another important reason to understand expectation is that
it gives you a sense of equanimity toward winning or losing
a bet: When you make a good bet or a good fold, you will know
that you have earned or saved a specific amount which a lesser
player would not have earned or saved. It is much harder to
make that fold if you are upset because your hand was outdrawn.
However, the money you save by folding instead of calling
adds to your winnings for the night or for the month. I actually
derive pleasure from making a good fold even though I have
lost the pot.
Just remember that if the hands were reversed, your opponent
would call you, and as we shall see when we discuss the Fundamental
Theorem of Poker in the next page, this is one of your edges.
You should be happy when it occurs. You should even derive
satisfaction from a losing session when you know that other
players would have lost much more with your cards
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