Poker Math That Matters

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only the hands that beat us. Our bet never gets called by a worse
hand. Say we bet $10 on the river.
0.64($0) + 0.36(-$10) = EV
$0 - $3.60 = (-$3.60)
Checking is certainly better than any of our betting options.
Let's go one step further with this hand and say our assumptions
have changed, and he would call with his 89. This certainly isn't
logical if he'll fold 99, but let's go with it for the sake of this
example. We can analyze his range: 89 represents 43% of his
range, 99 represents 21%, and the suited hands are 36%.
0.21($0) + 0.43($10) + 0.36(-$10) = EV
$0 + $4.30 - $3.60 = $0.70
We have a +EV bet because he's calling with more than 50%
worse hands than better hands. However, there are other things
to consider.
Against aggressive opponents, we may have to consider the
impact of being raised as a bluff. If we fold to his raise, we'll
need to have him call a bit more than 50% of the time to make
up for the times we fold to a worse hand. It's a good exercise to
examine different bet sizes when you're working on a situation
away from the table. Maybe you work out three or four different
bet sizes and examine them. You may have different
assumptions for each bet size. Sometimes you'll be surprised at
what you find. For example, let's say you have the nuts on the
river. Betting large with the nuts is normally a good plan.
However, if you're against an aggressive opponent, you may find
that a smaller bet might induce a raise from a large portion of his
hands that he'd normally fold to a large bet. The same is true