A poker hand missing a whole suit

Question

You are dealt 5 cards from a well-shuffled standard 52-card deck. What is the probability that at least one of the four suits is completely missing from your hand?

Accepted Answer

"At least one suit missing" is a union of four overlapping events (hearts missing, spades missing, and so on), so use inclusion-exclusion. Count hands with a given set of suits absent, using \binom{52}{5} = 2598960 total hands. - One particular suit absent: draw all 5 from the other 39 cards, \binom{39}{5} = 575757 hands. There are \binom{4}{1} = 4 suits to be the missing one. - Two particular suits absent: draw from the other 26, \binom{26}{5} = 65780. There are \binom{4}{2} = 6 such pairs. - Three particular suits absent: draw from the other 13, \binom{13}{5} = 1287. There are \binom{4}{3} …

A poker hand missing a whole suit

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