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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.

Stephen
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  • if you pull a side from another thread at the beginning and you knot it with the side you already have then you have only a knot outside. My question is : in the next try, if you pull another side of a thread what should you do ? the problem description seems a little bit ambiguous. – mounir ben salem Apr 30 '17 at 16:00
  • You put it back in the bag. So you essentially get a longer thread (come to think of it). And so the number of threads in the bag drops to $n-1$. – Stephen Apr 30 '17 at 16:09
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    Look here. – Jens Apr 30 '17 at 16:34
  • Awesome. This does answer the question. – Stephen Apr 30 '17 at 17:10

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