826. Birds on a Wire

Consider a wire of length 1 unit between two posts. Every morning $n$$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)$$F(n)$ to be the expected length of the wire that is painted. You are given $F(3)=0.5$$F(3) = 0.5$.

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

826. 电线上的鸟群

考虑两杆间一条长为 1 单位长度的电线。每天早晨，$n$$n$ 只鸟随机降落其上，电线上每一点等概率地被鸟占据。随后，对于每一只鸟，找到与其最近的鸟，并将两只鸟之间的电线涂色。

记 $F(n)$$F(n)$ 为有 $n$$n$ 只鸟时，被涂色的电线的期望长度。已知：$F(3)=0.5$$F(3) = 0.5$。

求所有 $F(n)$$F(n)$ 的均值，其中 $n$$n$ 取遍所有小于 ${10}^6$$10^6$ 的奇质数。将答案四舍五入到小数点后第 10 位。

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