Shuffled deck, expected position of the first ace

Question

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

Accepted Answer

A brute-force approach would be painful, but a symmetry argument makes this elegant. Set up the picture. Ignore everything except which cards are aces. There are 4 aces and 48 non-aces. Imagine laying the 48 non-aces in a row first: they create 5 "gaps" where aces can fall, one before all of them, three between them, and one after them (5 gaps because 4 aces split the line into 5 sections). By symmetry, the 4 aces are equally likely to land in any of these 5 gaps, so on average each gap holds the same number of non-aces. Count the non-aces before the first ace. The 48 non-aces are spread even…

Shuffled deck, expected position of the first ace

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