Birds on a Wire

By Mark Huber

Here's another way of thinking about it. Suppose that for a bird at position a on the wire we assign a random variable L_a that is the length of the yellow line to the left of a. So if a is closest to its bird on the right, and if the bird to the left of a is closer to its left bird, L_a = 0, otherwise it's some positive number.

What is the probability that L_a is in some tiny little interval around h? Or in probability notation, P(L_a in dh) = ?

There are two ways L_a can be close to h. One case is there is a bird at a - h, no bird in the interval (a - h, a), and no bird in (a, a + h). This occurs with probabilitty (1 - 2h)^{n-2} n dh.

Another case is when there is a bird at a - h, no bird from (a - 2h, a), and at least one bird in (a, a + h). This occurs with probability ((1 - 2h)^{{n -2 }} - (1 - 3h)^{{n - 2}}) n dh.

To find the expected value of L_a, this probability has to be multiplied by h, then integrated for h from 0 to 1/2. This gives an expected value of about (7/18)(1/n). Since there are n birds, the total yellow line is then 7/18 in expectation.

A variant of the Strong Law of Large Numbers then completes the proof that as the number of birds goes to infinity, the amount of line colored yellow is 7/18.

Geometric Probability

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