Odds Against an Open Raiser

In a tight, shorthanded hold'em game, an opponent well known to you raises from under the gun. You look down and find A-Q offsuit. Should you play? If your opponent had been forced all in on the blind, this would be an easy calculation, since we can just look up your win percentage against a random hand (64.4 percent, an easy call). But here, your opponent probably raised for some reason. Depending on the situation, you may be able to draw relatively strong inferences about his hand based just on this information.

Calculating your equity when your opponent holds a nonrandom hand is a problem we face constantly in poker. It's a hard problem to solve, even away from the table, because getting a good estimate may depend on a large number of variables that are more a matter of judgment than calculation, such as your opponent's standards for raising at exactly that moment. The more you can take into account, the more accurate your estimate will be. But you do need some kind of bare minimum in order to make good decisions.

We can chip away at the problem by decomposing it into manageable parts, and making some simplifying assumptions. In this case, we can break the problem down into two parts: calculating the relative likelihoods of all of your opponent's possible hands, and calculating your expectation in each case. These are not simple problems, either, but at least they take us a step closer to numbers we can actually use. For now, I'd like to punt on the second problem by assuming that for whatever reason, you just want to know your showdown win percentage (perhaps there will be no further action before showdown). What's left is just attaching percentages to all of the hands your opponent might hold.

Calculating the probability of a given hand, given your opponent's raise, is a straightforward application of Bayes Theorem, a well-known statistical theorem that has widespread applications in poker theory. In order to use Bayes Theorem here, you need two things. First, you need the prior probability of each hand, which is just the probability of the hand being dealt out of the 50 cards remaining in the deck. Second, you need the probability of your opponent raising for each possible hand.

The second, of course, is liable to be more difficult to come by, especially if you don't know your opponent well. But the more reasonable your guesses, the more reasonable your result will be. And you wouldn't be playing poker if you didn't think you could read minds, right? It's easy to understand how this works at the extremes. For example, if your opponent raises freely and often, regardless of his cards, you can pretty much ignore the raise, and go with the prior probabilities. If your opponent would raise with only A-A or K-K, the posterior probabilities will be drastically different from the prior probabilities. While these two examples are straightforward, there is plenty of room for gray area, especially when your opponent's raising habits are unusual, or when they're affected greatly by circumstances (especially in a tournament).

I'm not going to say much in this column about how these numbers are calculated, except to point out that it is a straightforward application of Bayes Theorem, and that you can look it up with a simple web search. I've carried out the actual calculations using a computer spreadsheet program, which allows me to punch in different possibilities to see how they affect the outcome. In case you'd like to see exactly how these numbers are derived, you can get a copy of the spreadsheet from my website: http://www.seriouspoker.com/spreadsheets/.

As is the case with many other analysis situations in poker, it's impossible to do these calculations in your head at the table, and it's impossible to memorize all possible combinations. So, the best solution may be to run the numbers for combinations of interest, and try to reach the point where your intuition is good enough that you can guess the answer before you see it. I have space here for only a few examples of how these calculations turn out for different situations. But if you'd like to get a feel for the numbers yourself, playing with the spreadsheet might be a fun and helpful exercise.

So, let's visit the situation I opened with: You hold A-Q offsuit against an open raiser. Suppose this opponent will raise with only five hands. The table below shows you the posterior probability of each possible hand, as well as your percentage equity in the pot. The second column is just the frequency of making each hand among the 1,225 possible hands. I find frequencies a little more intuitive, so I've included them instead of percentages. The third column is your estimate of your opponent's raise percentage. The fourth column is the posterior probability of each hand, calculated using Bayes Theorem. The fifth column contains your win percentage for that hand, calculated exactly by iterating all possible boards (just to make things easy on myself, I've made assumptions about suit overlap, so the win percentages might be very slightly off). And the final column contains the weighted win percentage, which is just the probability of your opponent holding that hand and your winning. The sum of the percentages in the final column is your percentage equity in the pot, should you reach showdown with no more information.

So, without knowing exactly which hand your opponent holds, we can say the most likely single hand is K-K, followed closely by A-K. And your chances of winning a showdown look to be about 23.5 percent, meaning you need pot odds of about 3.25-to-1 to call (again, assuming your call will close the action). Since all five of these hands are favorites against A-Q, this may not be the most difficult decision. But if someone is all in, you could have good enough pot odds for the call. So, knowing your win percentage, given the range of your opponent's possible hands, should be valuable, even when it's clear you're at best a moderate underdog. A more obvious example occurs when your tight opponent occasionally bluffs. Adding a single row to the table for the remaining hands changes things considerably:

I've fudged a bit on the win percentage for "Other" by using the showdown percentage against a random hand, which should be slightly pessimistic (given that we've already accounted for five tough hands). Nonetheless, we see that your showdown win percentage has vaulted from 22 percent to more than 50 percent, all because once in 33 rounds, your opponent will run a stone-cold bluff from under the gun. It may seem counterintuitive that such a small bluff percentage should affect the numbers this way, but that just reflects the fact that there are so many more garbage hands than real hands. Bayes Theorem doesn't make the same mistake that people often do with their intuitions, underestimating the impact of base rates. Of course, depending on the nature of the game (limit or no-limit, shorthanded vs. full, ring game vs. tournament, and so on), a bluff rate of 3 percent may be either maniacal or excessively tight. The numbers above are just one example.

Finally, let's see what happens if your opponent avoids the stone-cold bluff completely, but occasionally takes a shot with medium pocket pairs and a few other hands.

Now you need about 2-to-1 odds to call, which is definitely worth knowing. I didn't fudge the numbers to make it come out even, but there's still lots of room to quibble. Only you know who's sitting across from you at a given moment, so these are really just illustrative examples. But you can learn a few things just from these examples. First, while A-Q is anywhere from a moderate to a huge underdog against a tight raiser, and may be liable to pay off more bets after the flop, 3.25-to-1 isn't so long that it's an automatic fold under all possible circumstances. Second, it doesn't take a wild and crazy bluffer to boost your equity to the point where you're a small favorite. Even if your opponent is otherwise very tight, a bluff frequency as low as once in 30 opportunities can make your call or raise worthwhile.

It would be easy to fill a volume with these kinds of tables, not to mention minute interpretation of all of the numbers they include, but three tables and a brief tour is a good starting point. Of course, in practice, knowing the distribution of posterior probabilities for your opponent's holecards won't be all the information you need, especially if both of you have substantial stacks. But, understanding this general approach for assessing your opponent's holdings is a critical starting point to understanding how to adjust your play when your opponent's action is informative.

Editor's note: Daniel Kimberg is the author of Serious Poker, and he maintains a poker web site at www.seriouspoker.com.