Pairs in poker

An article by Y. L. Cheung [1] gives reasons why poker is usually played with five cards. The author gives several reasons, but here I'll just look at one reason: pairs don't act like you might expect if you have more than five cards.

In five-card poker, the more pairs the better. Better here means less likely. One pair is better than no pair, and two pairs is better than one pair. But in six-card or seven-card poker, a hand with no pair is less likely than a hand with one pair.

For a five-card hand, the probabilities of 0, 1, or 2 pair are 0.5012, 0.4226, and 0.0475 respectively.

For a six-card hand, the same probabilities are 0.3431, 0.4855, and 0.1214.

For a seven-card hand, the probabilities are 0.2091, 0.4728, and 0.2216.

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[1] Y. L. Cheung. Why Poker Is Played with Five Cards. The Mathematical Gazette, Dec., 1989, Vol. 73, No. 466 (Dec., 1989), pp. 313-315

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