Solution & Explanation

The problem at hand involves determining the expected number of loops formed when randomly tying ends of ropes together. Let's break down the solution step-by-step:

Problem Setup

• N ropes are in the bucket, each with 2 ends, resulting in a total of 2N ends.
• The process involves repeatedly selecting two ends and tying them together.
• A loop is formed when the two ends of the same rope are tied together.

Probability of Forming a Loop

1. Initial Selection:

   • Pick one end at random. There are (2N - 1) remaining ends.
   • The probability that the next selected end is the other end of the same rope (thus forming a loop) is 1/(2N - 1).

2. Subsequent Selections:

   • After forming a loop, there are N - 1 ropes left, and 2(N - 1) ends.
   • The probability of forming another loop in the next selection is 1/(2N - 3), and this pattern continues.

Recursive Formula

The expected number of loops E(N) can be defined recursively:

• Base Case: If there is only 1 rope (N = 1), the expected number of loops is 1 because the only two ends will be tied together.

• Recursive Case: For N > 1, the expected number of loops is given by:

  $E(N) = \frac{1}{2N-1} + E(N-1)$

  This formula accounts for the probability of forming a loop in the current selection and the expected loops from the remaining ropes.

Solving the Recursive Formula

The recursive formula can be expanded into a summation:

$E(N) = \sum_{k=1}^{N} \frac{1}{2k-1}$

This represents the expected number of loops as the sum of probabilities of forming loops at each step, from 1 to N.

Explanation

• Conceptual Understanding: Every time two ends are tied together, there's a chance to form either a loop or a longer rope. The probability of forming a loop decreases as more ends are tied, which is captured by the decreasing fraction $1/(2N - 1)$.
• Expected Value: The summation of probabilities provides a comprehensive measure of the expected number of loops, considering all possible outcomes as ropes are tied.

Conclusion

The expected number of loops formed when randomly tying ends of ropes is calculated by summing the probabilities of forming loops at each step, represented by the formula:

$E(N) = \sum_{k=1}^{N} \frac{1}{2k-1}$

This approach effectively models the stochastic process of tying ends and offers a clear method to calculate expected loops for any given number of ropes N.