Expected position of the first spade

Question

A standard 52-card deck is shuffled uniformly, and you turn cards over one at a time. What is the expected position of the first spade?

Accepted Answer

The 13 spades divide the 39 non-spades into 13 + 1 = 14 gaps: before the first spade, between consecutive spades, and after the last. By symmetry each non-spade is equally likely to fall in any of the 14 gaps, so each gap holds \frac{39}{14} cards in expectation. The first spade sits just after the first gap: E[	ext{position}] = \frac{39}{14} + 1 = \frac{53}{14} \approx 3.79. Indicator check. A specific non-spade c precedes the first spade iff, among c and the 13 spades (14 cards total), c comes first, probability \frac{1}{14}. So the expected number of non-spades before the first spade is 3…

Expected position of the first spade

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