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 probability $(1 - 2h)^{{n-2}}$ 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 $\displaystyle \frac{1}{2}.$ This gives an expected value of about $\displaystyle \frac{7}{18}\cdot\frac{1}{n}.$ Since there are $n$ birds, the total yellow line is then $\displaystyle \frac{7}{18}$ 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 $\displaystyle \frac{7}{18}.$

Geometric Probability

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