What is the probability that a five-card poker hand contains two pairs (that is, two of each of two different kinds and a fifth card of a third kind)?
Answers
Answer: Two pairs [one pair of each two different face values and a card of
a third face value]
13C2 x 4C2 x 4C2 x 11 x 4
Prob(2 pairs) = ------------------------- = .0475390
52C5
The various terms in above expression arise as follows.
13C2 = (13 choose 2) is the number of ways of choosing the two
different face values for the two pairs.
4C2 = (4 choose 2) is the number of ways of choosing the two suits for
one pair, and the second 4C2 is the number of ways of choosing the two
suits for the second pair.
11 is the number of ways of choosing the face value for the 5th card.
It must not be the same face value as either of the pairs.
4 is the number of ways of choosing the suit for the 5th card.
52C5 = (52 choose 5) is the number of unrestricted ways that 5 cards
can be selected from the 52 in the pack.
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