Compute the expectation of the number of resulting loops

Question

There are n strings, which of course have 2n ends. Then randomly pair the ends and tie together each pair. Let r.v. L be the number of resulting loops. Compute E[L].

Answer 1

When we tie the first end we have a chance of $\frac{1}{2n-1}$ of creating a closed loop and losing two ends and a chance of $\frac{2n-2}{2n-1}$ of not creating a closed loop, but losing two ends anyway.

Hence $E_n(L) = \frac{1}{2n-1}+E_{n-1}(L)$ and thus $E_n(L) = \sum_{i=1}^n\frac{1}{2i-1}$