Question: Suppose we turn over cards simultaneously from two well shuffled decks of ordinary playing cards. We say we obtain an exact match on a particular turn if the same card appears from each deck: for example, the queen of spades against the queen of spades. Let pm equal the probability of at least one exact match.
pm= 1-(1/2!)+ (1/3!) - (1/4!) +... - (1/52!).
Hint: Let Ci denote the event of an exact match on the ith turn. Then pm= P(C1 U C2 U...U C52). Now use the general inclusion-exclusion formula. Note that: P(Ci)= 1/52 and hence p1= 52(1/52) = 1. Also P(Ci intersect Cj)= 50!/52! and, hence, p2= (52 choose 2)/ (52*51).
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