You’re given a bag containing $n$ pieces of thread of length $l$.  $ $ You’re asked to reach down the bag and randomly pull out one end of a thread.  You’re then asked to knot this end to another end of a thread (that you also pull out randomly from the bag).  $ $  $ $ You have to do this $n$ times.    At the end of this activity, how many loops of thread will you have?  $$ $$    I was given this problem some time back, but sadly, it didn’t come with a solution. After multiple approaches, I’m fairly sure that we can’t give a definite answer as a function of $n$ and/or $l$ (at least, it doesn’t seem obvious). $ $   It seems the answer will be more probabilistic and less combinatorial $ $ (I may be wrong).   e.g.: the probability of making a loop on the first two picks would be $\frac{1}{2n-1}$, on the next two picks (if you did make a loop on the first two) would be $\frac{1}{2n-3}$, and so on.  $ $  $ $ I’m not sure how to develop much further from here.
There are 100 ropes in a bag. In each step, two rope ends are picked at random, tied together and put back into a bag. The process is repeated until there are no free ends. What is the expected number of loops at the end of the process?
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