The Absent-Minded Secretary

What this permutation and indicator-variable question tests

This is a classic medium-difficulty probability problem that quant interviewers use to test whether you can compute expectation and variance elegantly using linearity of expectation and indicator random variables—rather than brute-force enumeration. The setup is simple, but the solution technique is powerful and widely applicable.

The problem rewards candidates who recognize that the total count can be decomposed into independent binary indicators (one per letter), each with the same success probability. From there, computing the expectation becomes straightforward; the variance calculation requires careful attention to covariance terms between indicators, which reveals whether you understand dependencies in non-independent indicator sums.

• Indicator random variables and linearity of expectation
• Variance of a sum: expansion and covariance terms
• Permutation symmetry and marginal probability
• Derangements and fixed points