A pair or better in five cards
Five cards are dealt from a well-shuffled standard 52-card deck. Find the probability that at least two of them have the same rank.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Use the complement: all five ranks different.
Choose the five ranks, then a suit for each.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. It is easier to count hands with five different ranks and subtract.
Step 1: five different ranks. Choose the ranks in ways and a suit for each card in ways: hands.
Step 2: all hands. .
Step 3.
So a five-card hand has at least a pair (pair, two pairs, three or four of a kind, full house) almost half the time. The other half includes the straights and flushes with five distinct ranks.