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EXAMPLE 1 - Limit Hold'em

In poker, expected value still defines the average outcome when taking into account the
probability
of certain events.
With so many situations where probability dictates the outcome, it stands to reason that the
concept of expected value will apply a lot in poker situations. To illustrate this,
let's
look at a simple limit Hold'em example:

To answer this, we'll first have to determine how often he calls, how often he raises,
and how we fare against each of those. Let's assume that we somehow know his exact
range and it is only Ax, where the non-ace card is a club. We think that A9, A7, and A6
will call, and A5, A4, and A2 will fold. AQ with the Qc or AK with the Kc will make it
two betting rounds.

Let's look at our EV formula, and state that situation X is the one where we bet, he
raises, and we fold. Situation Y is where we bet and he calls. Situation Z is where
we bet and he folds.

We know that each of his possible poker hands is equally likely (2 aces left, times the specific
other card means two combos each), and there are 8 total hands he can have. So we can quickly
determine the probability that he holds each type of hand.

When we bet and get raised, we have lost our $10 bet. When we bet and get called, we make the
$100 in the pot, plus the $10 from the call. When we bet and he folds, we just win the $100 in
the middle.

amt(X) = -$10

amt(Y) = $110

amt(Z) = $100

EV(bet) = P(X)amt(X) + P(Y)amt(Y) + P(Z)amt(Z)

EV(bet) = $2.50 + $41.25 + $37.50

This seems great right away - but to know whether it's better than checking, we must also
determine
the EV of checking.

The probability is a little different this time because there are only two outcomes - we win at
showdown, or lose.

The payouts are different too. When we win, we win the $100 in the pot, but when we lose, we
don't
lose anything additional.

So putting this all back together, we can find the EV of checking:

EV(check) = P(win)amt(win) + P(lose)amt(lose)

We find that this is actually fairly close. The difference in EV of betting as compared to
checking
is only $1.25, or 1/8 of a big bet. This is not a huge difference so, if we have some
uncertainty
in our reads about his range, it could be a spot where a check would be preferable.

For instance, if he actually raises the 9c but we don't think he does, we may fold incorrectly
and
lose the pot too often, decreasing the EV of a bet below the EV of a check.

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EXAMPLE 2 - No Limit Hold'em

Let's take a look at an example from no limit Hold'em. We're on the river and we hold J9dd on a
board of Ad Kd 4c 7h 8s, for nothing but a busted flush draw. Our opponent has checked to us,
and
we are deciding whether or not to bluff all in. The pot is $200, and we have $100 in our stack.

Due to how the hand played out, we think our opponent can't have sets or flopped two pair, but
he
could hold AQ-AT, QTdd, QQ-99.

We think that if we shove, he will fold QTdd, QQ
through 99, but will call with his Ax hands.

Let's look at the number of each of those hands and the resulting probability that he will call a shove:

QQ-99

18 combinations

(we block Jd and 9d)

EV(bet) = $2.50 + $41.25 + $37.50

If we shove and he folds, we win the $200. If we shove and he calls, we are always beat and we
lose
our $100 stack.

amt(call) = -$100

amt(fold) = $200

EV(shove) = P(call)amt(call) v P(fold)amt(fold)

EV(shove) = -$65.45 v $69.10

EV(shove) = $3.65 difference

The expected value of a check is actually $0 here because we never win the pot when we check it
behind.

We see that shoving is slightly preferable to checking, even though we get called well over 50%
of the time.

This is a situation that can arise fairly often in poker - since most bets are for a pot-sized
bet or less even in no limit games, you are typically laying yourself a price of better than
1:1

In this case, we lay 2:1 by risking a bet worth half the pot, so we only need it to succeed 1/3
of the time to be break even as compared to checking.