Loops from tying noodles

A bowl holds 100100 noodles, each with two ends (200200 ends total). You repeatedly pick two ends uniformly at random and tie them together, until no free ends remain. On average, how many closed loops form? Round to the nearest whole number.

Show hints (2)+
  1. With kk strands, tying a chosen end closes a loop with probability 1/(2k1)1/(2k-1) (the strand's own other end).
  2. Sum by linearity: k=11001/(2k1)3.28\sum_{k=1}^{100} 1/(2k-1) \approx 3.28.

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Asked at: Jane Street, Two Sigma

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