Consider a loop of string of unit length. Suppose we cut the string independently and at random in n places. This will divide the loop into n pieces.
1. What is the expected (average) size of the smallest piece?
2. What is the expected (average) size of the largest piece?
If you can't find an exact answer the asymptotic behaviour (to leading order) as n goes to infinity will suffice.
Update (11/12/09): Solution in comments!!