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When you can't actually put a person on a hand but have reduced his possible hands to a limited number, you try to use mathematics to determine the chances of his having certain hands rather than others. Then you decide what kind of hand you must have to continue playing. Using mathematics is particularly important in draw poker, where your main clue to what an opponent might have is what you know about his opening, calling, and raising requirements.

If, for example, you know an opponent will raise you with three 2s or better before the draw, you can resort to mathematics to determine what hand is favored to have him beat. It works out to something like three queens. Obviously, then, if you have three 3s, it's not worth calling that opponent's raise on the chance that he has specifically three 2s. But if you have something like three Ss or three 6s, the pot odds make it correct to call because now not only might you draw out on a better hand by making a full house or four-of-a-kind, but there are a few hands your opponent could have which you already have beat.

Sometimes you can use a mathematical procedure based on Bayes' Theorem to determine the chances an opponent has one or another hand. After deciding upon the kinds of hands your opponent would be betting in a particular situation, you determine the probability of his holding each of those hands. Then you compare those probabilities. If, for instance, in draw poker you know a particular player will open either with three-of-a-kind or two pair but will not open with one pair and will check as a slow play with a pat hand, then it is 5-to-2 against that player's having trips when he does open. Why is this so? On average, according to draw poker distribution, a player will be dealt two pair 5 percent of the time and trips 2 percent of the time. When you compare these two percentages, you arrive at a ratio of 5-to-2. Therefore, the player is a 5-to-2 favorite to have two pair.

Let's say in hold 'em an opponent puts in a big raise before the flop, and you read him for the type of player who will raise only with two aces, two kings, or ace, king. The probability that a player gets two aces on the first two cards is 0.45 percent. The probability of his getting two kings is also 0.45 percent. So he will get two aces or two kings 0.9 percent of the time on average. The probability of his getting an ace, king is 1.2 percent. By comparing these two probabilities -1.2 percent and.9 percent you deduce that the chances are 4-to-3 in favor of your opponent's having ace, king rather than two aces or two kings. Of course, knowing your opponent is a 4-to-3 favorite to have ace, king is not enough by itself to justify calling his raise with, say, two queens. You are a small favorite if he does have ace, king, but you're a big underdog if he has two aces or two kings. Nevertheless, the more you know about the chances of an opponent's having one hand rather than another when he bets or raises, the easier it is for you to decide whether to fold, call, or raise.

It does not, of course, take a mathematical genius to realize that another king on the table decreases the chances of an opponent's having two kings before he raises, so using math to read hands does not always require the precise knowledge of card-distribution probabilities presented here. Furthermore, you need to complement mathematical conclusions with what you know about a player. For example, in a relatively small-ante games, some players might not raise with two kings when there is no other king showing in hopes of making a big hand, but they will raise with two kings when there is a king showing to try to win the pot right there. They decide to go for the pot right away precisely because of the presence of that other king, which reduces their chances of improving. When you are up against such players, the presence of another king might actually increase the probability of their having two kings after they raise - not on the basis of mathematics but on the basis of the action they have taken and what you know about the way they play.

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