Making Money With Bayes Theorem

In my last article, I introduced Bayes theorem, a basic concept in the study of probability. Understanding how Bayes theorem works in poker is critical to making many different types of decisions at the table. The example I gave last article was how to figure out if someone is bluffing frequently enough to justify a call. If you haven’t yet read that article, read it first before you continue with this one.

In this article I’ll discuss more generally the idea of making assumptions about how your opponents play and then tailoring your strategy to these assumptions. In many cases you should make and act on these assumptions before you’ve seen your opponent play a single hand. Bayes theorem underpins the logic that justifies this.

Let’s talk about continuation bets for a little bit. The typical $2-$5 player makes plenty of continuation bets. If you watch a $2-$5 game for a while, you’ll see that the preflop raiser will make a flop continuation bet in most pots where he gets only one caller. This is true whether the raiser is in or out of position, whether the caller is in the blinds or not, and almost regardless of flop texture. If someone raises preflop and gets one caller, he’s likely to bet the flop.

Now, if you understand no-limit hold’em more deeply, you’ll know that all of these factors I listed above should affect your continuation betting decision. You should bet more when you’re in position than when you’re out. You should bet more against callers from the blinds than against callers from outside the blinds. You should bet more on certain flop textures than on others—and which flop textures to bet depend on whether you’re in or out of position.

As a result, there are plenty of situations where you should continuation bet nearly 100 percent of your hands. But there are also situations where you should check most of your hands, even though you were the preflop raiser and you got only a single caller.

My assumption, however, is that most $2-$5 players don’t understand these nuances. Their continuation betting percentage is going to be mostly constant over all these variables. They decide whether to bet or not based on how their two cards connect with the flop and consider relatively little else. But certainly some $2-$5 players do understand these ideas. If you were to watch them for a while, you’d watch their continuation betting habits change based on the circumstances.

With that context, let’s look at a hand. It’s a $2-$5 game with $1,000 stacks. A player limps, and a player you’ve never played a hand against before makes it $25 to go from two off the button. You call on the button. The blinds fold, and so does the limper. There’s $62 in the pot, and you’re heads-up to the flop.

The flop is:

T♣ 9♠ 5♠

Your opponent bets $35.

In general, this is a bad situation for your opponent to continuation bet. He was called by a player not in the blinds, and he is out of position. Furthermore, this flop texture is dynamic, which favors the in position player. If there were ever a time to be cautious with continuation betting in a heads-up pot, this is it.

But your opponent bet. The question you have to answer for yourself is, “Is this bet a mistake?” The answer is not obvious. It could be a mistake, but it could be a perfectly legitimate bet with a hand like pocket aces. You’ve never played a hand with this fellow before, so you have no idea his personal tendencies.

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Let Bayes theorem be your guide. We know two things. First, we know that many $2-$5 players make errors in their continuation betting. Second, we know that this is a situation that calls for relatively little continuation betting. Given that your opponent did bet, how likely is it to be an error?

Let’s say that 80 percent of $2-$5 players fail to adjust their continuation betting strategy properly to the situation. I think the actual number is higher, but let’s go with 80 percent for this example. The other 20 percent of $2-$5 players will continuation bet perfectly—that is, the chance that their bet is an error is zero. (Again, this is a very generous assumption, since no one plays perfectly.)

Now let’s say that the proper continuation betting frequency in this particular scenario is 30 percent. These comprise the hands that are strong enough to bet despite the bad situation plus enough bluffs to balance. With the other 70 percent of hands, the preflop raiser should check.

So there are four possibilities. A “weak” $2-$5 player is one of the 80 percent who bets imperfectly. A “strong” player is one who is perfect. A “weak” hand is one of the 70 percent that shouldn’t be bet. A “strong” hand is one of the 30 percent that should be bet.

1. You could have a strong player with a strong hand.
2. You could have a strong player with a weak hand.
3. You could have a weak player with a strong hand.
4. You could have a weak player with a weak hand.

You can figure out how likely each of these scenarios is by multiplying probabilities. A strong player with a strong hand will occur 0.2 x 0.3 = 0.06 or 6 percent of the time. A strong player with a weak hand is 0.2 x 0.7 = 0.14 or 14 percent of the time.

A weak player with a strong hand is 0.8 x 0.3 = 0.24 or 24 percent of the time. A weak player with a weak hand is 0.8 x 0.7 = 0.56 or 56 percent of the time.

Only three of these scenarios result in a bet, however. A strong player with a weak hand doesn’t bet. The original question was, “Is this bet a mistake?” The bet is a mistake only when a weak player has a weak hand. Scenario 4 happens 56 percent of the time, so we know that it’s more likely than not that the bet was a mistake. But the answer to the original question isn’t 56 percent. We have to use Bayes theorem to obtain the right answer.

Say we played this hand out 100 times. In 14 of the cases (i.e., 14 percent of the time), your opponent would be strong with a weak hand, so there would be no bet. We must exclude these cases because your opponent did bet. Of the remaining 86 cases, 56 of these we consider an error. So the chance that the bet is an error is 56/86 = 0.651 or about 65 percent.

You’ve never seen this player before in your life, but you can estimate that this particular continuation bet is an error about 65 percent of the time. (We also assumed that a weak player would continuation bet 100 percent of hands. This is close to true for some players, but high for others.)

Your best defense in this situation is to assume that the player is bad and that the bet is in error. You could raise it immediately, or you could call, planning to evaluate your opponent’s turn behavior.

When you assume your opponent is bad, you will be wrong occasionally. But you’ll do better if you make the assumption and get burned sometimes than if you treat unknown opponents as if you know nothing about them or—even worse—as if they played perfectly.