“He Probably Has a Set”

If you’ve ever said or thought “my opponent probably has a set” you’re likely to have some holes in the way you read hands. I hope I can plug at least one of them today.

Reading hands isn’t about putting your opponent on a single hand. It’s about putting your opponent on a range of hands and narrowing that range as you get more information. Exclusion. Most people get this right; they start excluding things from their opponent’s likely holdings based on his actions. The sentence “it seems unlikely that he has AA since he didn’t re-raise preflop” is an example of this. We can surely discard a whole lot of unlikely holdings much the same way, especially after we see how they act on the flop.

This is the good way to figure out hands. Assign a range of possible hands, and discard holdings as the hand plays out. A good thing to keep in hand when trying to figure this out on the fly is that it’s the number of combinations that matter. For this reason, if you “know” that your opponent has either AA or AK, then he’s more likely to have AK because there are 16 ways he can have that hand, but only 12 ways he can have AA.

Now we take this step a bit further:

We trust that a particular player will only re-raise preflop with, say, JJ and up, AQ and AK. We have 88. We know that he will play strong hands straightforwardly, and is willing to move all the chips in with any hand as strong as, or stronger than, top pair-top kicker.

$1/$2 NLHE: We make it $6 from UTG with 88, and this particular player re-raises to $14 on the button. We know have his range down to JJ+, AK or AQ.

The flop comes A-Q-8. We check, he bets $20, we raise it to $50 and he moves all-in for the rest of his ($200) stack.

Should we call?

In this situation, he either has AA, AK, AQ or QQ. These are the only hands he will move in with. We crush some of his range, and we’re completely crushed by the rest of it. What’s our move? I know the answer before doing the calculations. Do you? Either way, here goes:

Given that there’s an ace and a queen on the board, some hands become less likely than others, “combinatorically.” Whereas there are 16 ways to get dealt AK from any two cards out of the full deck, one of the aces is now spoken for. There are now only 12 ways left to be dealt AK. Let’s look at all the hands he can have:

AK - 12 (4*3) ways.

AQ - 9 (3*3) ways. Both a queen and an ace are spoken for.

AA - 6 (3*2) ways.

QQ - 6 (3*2) ways.

There are 12+9+6+6 = 33 possible hands he can have been dealt preflop that make sense. We’re huge favorites against 22 of these, or precisely two-thirds. We must call.

What’s the point of this example? To illustrate how combinatorics make it so that certain hands become a lot more likely than others. And specifically, per the blog post title, that our opponent very rarely can be said to “probably have a set.” Any time that our opponent plays a hand strongly, when there is no flush or straight available, it’s a very rare thing for a set to be the most likely holding. Look for possible two-pair combinations or an overplayed top pair. If you ever find yourself folding a set because you think that your opponent must have a bigger set, you’re costing yourself money.