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Problem: If I pick five cards out of a standard 52-card deck, what's the probability that I have a pair (but not anything better)?

Solution:

    We'll solve in two parts

    (a) The total number of possible poker hands is:

      (52 choose 5) = (52 * 51 * 50 * 49 * 48) / (5 * 4 * 3 * 2 * 1)

      = 2,598,960

    (b) The total number of hands which have a pair (but nothing better) is:

      number of pairs * number of ways to pick 3 cards which don't match the pair, or each other

      = (13 * (4 choose 2)) * (48 * 44 * 40) / (3 * 2 * 1)

      = (13 * ((4 * 3) / (2 * 1)) * (48 * 44 * 40) / 6

      =1,098,240

      Note: 13 is A, 2, ...Q, K; (4 choose 2) picks the suits

    So the probability that a random hand has a pair (but nothing better) is:

      1,098,240 / 2,598,960

      = 0.4225690276 (about 42 percent)

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