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Letters in matching envelopes

ProbabilityEasy

A clerk puts $8$ different letters into $8$ addressed envelopes, one letter per envelope, with all assignments equally likely. What is the expected number of letters that land in their own correct envelope?

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Hint

Write the count as a sum of indicators and use linearity of expectation.

Solution

Let each letter have an indicator equal to $1$ if it lands in its own envelope and $0$ otherwise.
By symmetry each letter is in its own envelope with probability $\frac{1}{8}$ .
By linearity the expected count is $8 \times \frac{1}{8}$ .
The answer is $1$ .

Answer: $1$

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