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826. 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.