What's Your Starting Hand Really Worth?

Steve Brecher

Most players know that pre-flop position is important in hold 'em. The earlier your position, the more players there are behind you and, unless you hold pocket Aces, the bigger the chance that one of them will have a hand better than yours.

There is another aspect to position: It's better to act after your opponent(s) rather than before. But for this tip, I'm going to investigate the chances that a player behind you will have a better hand.

There is no universal definition of what "better" means when comparing hold 'em starting hands. For this article, I needed some reasonable, quantifiable criterion. So in the following, I'm assuming that one hand is "better" than another if its showdown equity is greater. A hand's showdown equity against another hand is the average portion of the pot it will win across all

possible combinations of board cards. This is similar to the percentages that TV poker programs display next to player hands when the players are all-in. If you're interested in investigating this for yourself, there are several free computer programs and websites which calculate the showdown equities of user-specified competing hands.

For example, Ah 2d all-in pre-flop against Kc Qc will, over all possible boards, win an average of 53.9% of the pot. So the A-2 is the "better" hand against K-Q suited by our definition. Obviously, it is not better for all purposes; at a full table I'd usually open-raise in early position with K-Q suited, but toss A-2 offsuit.

Given some specific hand category - such as K-Q suited - we'll need to know the chance that a random hand dealt from the remaining 50 cards will be "better." This requires that we have a showdown equity calculation for each of the 1,225 possible opposing hands and tabulate against how many of them the K-Q suited has the worse (less than 50%) equity. It turns out that 238

of the 1,225 possible opponent hands are "better" in this sense. So we say that the chance of a random hand being better than K-Q suited is 238/1,225 or 19.4%; conversely, the chance that a random hand will not be better is 80.6%. This tabulation would be too tedious to do by hand. For the example results below, I developed some simple software to do the calculations.

Suppose that you are considering an opening bet pre-flop. There are players yet to act behind you. I'll denote the number of hands to play behind you as N. For example, if you're on the button, then there are two hands - the blinds - behind you, and N would be equal to 2. What is the probability that none of some number of random hands will be better than yours? It is the

chance that one random hand will not be better than yours multiplied by itself N-1 times, which is the same as saying it's that probability raised to the Nth power. For example, if there's a 40% chance that a random hand won't be better (i.e., a 60% chance it will be better), then the chance that none of three random hands will be better is 40% x 40% x 40%, or 0.4 to the 3rd power, which equals 0.064. Hence, the chance that at least one of the three hands will be better is 1.0 - 0.064 or 0.936 or 94%.

I think the most interesting thing about these numbers is the difference between earlier and later positions. This is something to consider when you're thinking of open-raising in early position.

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