Mathematical
expectation is the amount a bet will average winning
or losing. It is an extremely important concept for
the gambler because it shows him how to evaluate most
gambling problems. Using mathematical expectation
is also the best way to analyze most online poker
games plays.
Let's say you are betting a friend $1, even money,
on the flip of a coin. Each time it comes up heads,
you Win; each time it comes up tails, you lose. The
odds of its coming up heads are 1to1, and you're
betting $1to$1. Therefore, your mathematical expectation
is precisely zero since you cannot expect, mathematically,
to be either ahead or behind after two flips or after
200 flips.
Your hourly rate is also zero. Hourly rate is the
amount of money you expect to win per hour. You might
be able to flip a coin 500 times an hour, but since
you are getting neither good nor bad odds, you will
neither earn nor lose money. From a serious gambler's
point of view, this betting proposition is not a bad
one. It's just a waste of time.
But let's say some imbecile is willing to bet $2 to
your $1 on the flip of the coin. Suddenly you have
a positive expectation of 50 cents per bet. Why 50
cents? On the average you will win one bet for every
bet you lose. You wager your first dollar and lose
$1; you wager your second and win $2. You have wagered
$1 twice, and you are $1 ahead. Each of these $1 bets
has earned 50 cents. 
Many
players will wait until the betsize doubles on the
river to raise with a strong hand. Also, many players
will semibluff if they pick up a flush draw on the
turn.
A semibluff is what used to be called raising with
outs. It's a certain kind of raise that's made before
the last round of betting, when there are more cards
to come, and there is some chance that the raise will
win the pot right there. Its a semibluff rather than
just a bluff if you also have some fallback, some
chance you'll end up winning the showdown anyway.
Raising with a flush draw is an example of a semibluff.
Basically, the idea is that you might win the pot
right now, but if you don't, all is not lost because
you might still improve to the best hand.
If the board looks like there is a chance your opponent
has a draw (such as having two hearts on the board)
then you need to always consider the possibility that
your opponent is semibluffing when he raises before
the last betting round.
