Recreational Math

Card and Dice Games

What are the probabilities and odds of being initially dealt certain hands in five-card poker and bridge?

Because there are a certain number of cards in a playing deck, mathematicians have worked out the probabilities and odds of being dealt certain hands for certain games. The following explains the odds in terms of “chances against : chances for.”

In five-card poker the probability of drawing a royal flush (a poker hand with the ace, king, queen, jack, and 10 all in the same suit) is 1.54 × 10-6, with odds of 649,739:1. For a straight flush (a poker hand with consecutive cards in the same suit, but not a royal flush), the probability is 1.39 × 10-5, with odds of 72,192.3:1. Three of a kind (three cards with the same value) has a probability of 0.0211, with odds of 46.3:1, while one pair (two cards with the same value) has a probability of 0.423, with odds of 1.366:1.

In bridge 13 top honors has a probability of 6.3 × 10-12 and odds of 158,753,389,899:1. A 12-card suit, ace high, has a probability of 2.72 × 10-9, and odds of 367,484,697.8:1. Getting four aces has a probability of 2.64 × 10-3, with odds of 377.6:1.


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