Shuffled deck, expected position of the second ace

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You shuffle a standard 52-card deck and turn cards face up one at a time from the top. On average, at what position does the second ace appear?

Accepted Answer

Use the same gap-symmetry picture, now tracking the second ace. Set up the picture. There are 4 aces and 48 non-aces. The 4 aces split the 48 non-aces into 5 equal gaps on average, each holding \frac{48}{5} = 9.6 non-aces. Count what comes before the second ace. Before the second ace sit: the first gap of non-aces, then the first ace, then the second gap of non-aces. That is two gaps of non-aces (each averaging 9.6) plus one ace already turned. So the number of cards before the second ace averages: 2 	imes 9.6 + 1 = 19.2 + 1 = 20.2. Position of the second ace. Add one for the second ace itse…

Shuffled deck, expected position of the second ace

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