Shuffled deck, expected position of the first spade

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 does the first spade appear?

Accepted Answer

Use the same symmetry idea as with aces, now with spades as the special cards. Set up the picture. There are 13 spades and 39 non-spades. Lay the 39 non-spades in a row: the 13 spades split that row into 14 gaps (one before, twelve between, one after). By symmetry the spades are equally likely to fall in any gap, so on average each gap holds the same number of non-spades. Non-spades before the first spade. The 39 non-spades spread evenly across 14 gaps, so the first gap holds on average: \frac{39}{14} 	ext{ non-spades.} Position of the first spade. Add one for the spade itself: EV = \frac{39…

Shuffled deck, expected position of the first spade

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