Expected loops from tying 100 noodle ends

A bowl contains $100$ noodles. Repeatedly, you pick two free noodle ends uniformly at random from all free ends and join them together, returning the result to the bowl. You stop once no free ends remain. What is the expected number of closed loops formed? Give the answer rounded to the nearest integer.

$3$. With $n$ noodles there are $2n$ ends; whichever end you pick first, its partner is that same noodle's other end with probability $\tfrac{1}{2n-1}$, closing a loop, and either way you are left with $n-1$ noodles. Hence the expectation is $\sum_{k=1}^{100}\tfrac{1}{2k-1}\approx 3.28$, which rounds to $3$.