1975 USAMO Problems/Problem 5

Problem

A deck of $n$ playing cards, which contains three aces, is shuffled at random (it is assumed that all possible card distributions are equally likely). The cards are then turned up one by one from the top until the second ace appears. Prove that the expected (average) number of cards to be turned up is $(n+1)/2$ .

Solution

We begin by induction. Our base case is naturally when $n = 3$ , as there can be no less than $3$ cards in the deck. The only way to turn up the second ace is to turn up the first, and then turn up the second, which requires $2$ moves. This indeed is equal to $(3 + 1)/2$ . Next, we assume that the expected number of cards required to pull the second ace from a deck of n cards is in fact $(n + 1)/2$ . Now, we see that to simulate the case with $n + 1$ cards, we simply must add a card to our deck.

1975 USAMO (Problems • Resources)
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1975 USAMO Problems/Problem 5

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