At least one suit missing from a 5-card hand

Question

Five cards are dealt at random from a standard 52-card deck (four suits of 13 cards each). What is the probability that at least one suit is completely missing from the hand?

Accepted Answer

Let As be the event that suit s is absent. We want P(A\clubsuit \cup A\diamondsuit \cup A\heartsuit \cup A\spadesuit), so we use inclusion-exclusion. Every count is over the \binom{52}{5} = 2{,}598{,}960 equally likely hands. - One suit absent: the five cards come from the other 39, so \binom{39}{5} = 575{,}757. There are \binom{4}{1}=4 suits. - Two suits absent: cards come from the remaining 26, so \binom{26}{5} = 65{,}780, with \binom{4}{2}=6 pairs. - Three suits absent: cards come from one suit of 13, so \binom{13}{5} = 1{,}287, with \binom{4}{3}=4 triples. - Four suits absent is impossibl…

At least one suit missing from a 5-card hand

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