Consider a wire of length 1 unit between two posts. Every morning 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 to be the expected length of the wire that is painted. You are given .
Find the average of where ranges through all odd prime less than a million. Give your answer rounded to 10 places after the decimal point.
考虑两杆间一条长为 1 单位长度的电线。每天早晨, 只鸟随机降落其上,电线上每一点等概率地被鸟占据。随后,找到每一只鸟与其最近的鸟,并将两只鸟之间的电线涂色。
记 为有 只鸟时,被涂色的电线的期望长度。已知:。
求所有 的均值,其中 取遍所有小于 的奇质数。将答案四舍五入到小数点后第 10 位。
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