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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:
(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:
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