## 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.