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Shuffled coats at checkroom

Expected valueEasy

$12$ traders leave their coats at a checkroom. The attendant hands the coats back in a uniformly random order, one coat to each trader. What is the expected number of traders who receive their own coat?

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Hint

Define an indicator for each trader and use linearity of expectation.

Solution

Let the indicator of trader $i$ equal $1$ if that trader gets their own coat, and $0$ otherwise.
By symmetry each trader receives any given coat with the same chance, so the probability of getting their own coat is $\frac{1}{12}$ .
By linearity of expectation the total is the sum of $12$ equal terms: $12 \times \frac{1}{12}$ .
The expected number is $1$ .

Answer: $1$

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