Imagine you have N pieces of rope in a bucket. You reach in and grab one end-piece, then reach in and grab another end-piece, and tie those two together. What is the expected value of the number of loops in the bucket?

Is the question and answer makes sense? I thought the answer is 1/(2n-1). I don't understand why the solution adds all probability from 1 to N case together? For the 2 ropes case, the p(1 loop) = 1/3. So expected number of loop is also 1/3, but why the answer is 1+1/3= 4/3?Am I missing something?

You are right, the long answer failed simple boundary condition: if you tie once after pick two end, the max number of loops is one! So the p(n) is [0,1], lol