A 13-card hand is dealt from a standard deck of 52 cards. What is
the probability that
(a) it contains no spades if it contains exactly 5 hearts,
P(no spades|exactly 5 hearts) =
P(no spades and exactly 5 hearts)
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P(exactly 5 hearts)
N(no spades and exactly 5 hearts)/C(52,13)
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N(exactly 5 hearts)/C(52,13)
Multiply top and bottom by C(52,13)
N(no spades and exactly 5 hearts)
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N(exactly 5 hearts)
N(no spades and exactly 5 hearts) =
(The number of ways to pick 8 from the 26 non-spades-non-hearts)
times
(the number of ways to pick exactly 5 hearts from the 13) =
C(26,8)·C(13,5)
N(8 non-hearts and exactly 5 hearts) =
(The number of ways to pick 8 nun-hearts from 39 non-hearts)
times
(the number of ways to pick exactly 8 hearts from the 13) =
C(39,8)·C(13,5)
Therefore,
P(no spades|exactly 5 hearts) =
C(26,8)·C(13,5) C(26,8)
--------------- = ------- = .0253930401
C(39,8)·C(13,5) C(39,8)
(b) it contains at least one spade if it contains exactly 5 hearts?
That is the probability of the complement event of part (a), so it's
1 - .0253930401 = .9746069599
Edwin