Generalizing a formula of Stanley, we prove combinatorially that the probability that \(1, 2, \dots , k\) are contained in the same cycle of a product of two random n-cycles is \(\frac{1}{k} + \frac{4 (-1)^n}{ \left( {\begin{array}{c}2k\\ k\end{array}}\right) } \sum _{\begin{array}{c} 1 \le i \le k-1 \\ i \not \equiv n \bmod 2 \end{array}} \left( {\begin{array}{c}2k-1\\ k+i\end{array}}\right) \left( \frac{1}{n+i+1} - \frac{1}{n-i}\right) .\)
