Mathematics

How Probability is used in Card Games



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Poker and bridge are not gambling games. Both contain an element of chance, but consistent winners and losers emerge in each game over time. In addition to their understanding of psychology, and their excellent memories, winners at bridge and poker employ a very precise knowledge of probability.

The chances of improving a poker hand in the draw are computable. The odds that a singleton king, or any other card, is to the left of dummy in bridge are knowable, especially after a round or two of bidding. Players who understand and take advantage of these odds will always outplay opponents who do not, even though they may not win every hand.

In bridge, you want to know who is about to run out of a suit, and, when possible, what specific cards each player is holding. After the bidding, one fourth of the cards will be face up on the table (the dummy hand), so this task is not as difficult as it might seem. In addition, you know what cards you hold, and how the bidding went. Possibly, you have some ideas about the personalities of the players, the way they bid, and the way they tend to play a hand.

Let's say you hold seven of the heart cards, and you can see four of them face up on the table in the dummy. Since there are 13 cards in a suit, you know that the two players whose hands you can't see hold the two missing heart cards. You know that these cards are the king and four of hearts, because those are the only cards you cannot see. The player on your left has two hearts, one heart, or no hearts. If you know probability, you know that the chances of that player having one heart are slightly better than fifty-fifty. The chances of that heart being the king are half of that.

These odds change with the number of hearts that you know about. Nevertheless, with a reasonably strong understanding of probability, it is possible to act upon reasonable certainties about the locations of important cards. In bridge, of course, a player will have also gathered valuable information from the bidding and from previous play.

It is not necessary to be able to understand Bayes' rule or apply Chebyshev's theorem in order to use probability. Tables of probabilities exist, and can be found in books on card games, and on the web at sites like Durango Bill's. Study is involved, of course.

This does not mean that players who know the odds will always win. It means that in order to win, card players must gain some knowledge of probability. Players who act upon knowledge rather than hunch will tend, over time, to come out way ahead.

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