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.