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Problem 826

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Birds on a Wire

Consider a wire of length $1$ unit between two posts. Every morning $n$ birds land on it randomly with every point on the wire equally likely to host a bird. The interval from each bird to its closest neighbour is then painted.

Define $F(n)$ to be the expected length of the wire that is painted. You are given $F(3) = 0.5$.

Find the average of $F(n)$ where $n$ ranges through all odd prime less than a million. Give your answer rounded to $10$ places after the decimal point.

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电线上的小鸟

两根电线杆之间有长为$1$的电线。每天早上，$n$只小鸟均匀随机地落在这段电线上。将每只小鸟和其最近邻居之间的电线染上颜色。

记$F(n)$为被染色电线的总长度的期望值。已知$F(3) = 0.5$。

当$n$取遍所有小于一百万的奇素数时，求所有$F(n)$的平均值。将你的答案四舍五入至小数点后$10$位。

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