Torpids
Problem 597
Suppose that, in a particular race, each boat rows at a steady speed log metres per second, where the are chosen randomly (with uniform distribution) between 0 and 1, independently from one another. These speeds are relative to the riverbank: you may disregard the flow of the river.
Let be the probability that the new order is an even permutation of the starting order, when there are boats in the division and is the course length.
For example, with and , labelling the boats as ,, in starting order with highest, the different possible outcomes of the race are as follows:
Therefore, .
You are also given that , rounded to 10 digits after the decimal point.
Find rounded to 10 digits after the decimal point.