Two shuffled decks: how many positions match?

Question

You have two identical 40-card decks. You shuffle each one thoroughly and independently, then lay them side by side and compare position by position: position 1 of deck A against position 1 of deck B, position 2 against position 2, and so on through position 40.

On average, how many of the 40 positions show the exact same card in both decks?

Accepted Answer

Comparing two random decks is the same as fixing one deck and randomly permuting the other, so this is a matching problem in disguise. Handle it one position at a time. Look at a single position. Whatever card deck A shows there, deck B's card in that slot is equally likely to be any of its 40 cards, so it matches with probability 	frac{1}{40}. Let Xi = 1 if position i agrees in both decks. Then E[Xi] = 	frac{1}{40}. Add over all 40 positions, X = X1 + \cdots + X{40}, and by linearity E[X] = 40 	imes \frac{1}{40} = 1. Independence of the two shuffles does not even matter here; all that mat…

Two shuffled decks: how many positions match?

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