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The problem of tying ropes together randomly until no loose ends remain is a fascinating exercise in probability and combinatorics. The goal is to determine the expected number of loops formed when all ropes are tied.
Probability of Forming a Loop:
At any given step with N
ropes, the probability of selecting two ends from the same rope (forming a loop) is given by:
P(loop)=2N−11
Conversely, the probability of selecting ends from different ropes (not forming a loop) is:
P(no loop)=1−2N−11=2N−12N−2
Expected Number of Loops:
As each loop reduces the number of ropes by one, the expected number of loops formed can be calculated by summing the probabilities over all steps:
E(loops)=∑N=11002N−11
This sum approximates to about 3.284 loops when calculated numerically.
To verify this result, a simulation can be conducted:
The expected number of loops formed when all ropes are tied is approximately 3.28. This result aligns with both theoretical calculations and empirical simulations, demonstrating the elegance of probability in random processes.