Unfair race
Problem 573
runners in very different training states want to compete in a race. Each one of them is given a different starting number according to his (constant) individual racing speed being .
In order to give the slower runners a chance to win the race, different starting positions are chosen randomly (with uniform distribution) and independently from each other within the racing track of length . After this, the starting position nearest to the goal is assigned to runner , the next nearest starting position to runner and so on, until finally the starting position furthest away from the goal is assigned to runner . The winner of the race is the runner who reaches the goal first.
In order to give the slower runners a chance to win the race, different starting positions are chosen randomly (with uniform distribution) and independently from each other within the racing track of length . After this, the starting position nearest to the goal is assigned to runner , the next nearest starting position to runner and so on, until finally the starting position furthest away from the goal is assigned to runner . The winner of the race is the runner who reaches the goal first.
Interestingly, the expected running time for the winner is , independently of the number of runners. Moreover, while it can be shown that all runners will have the same expected running time of , the race is still unfair, since the winning chances may differ significantly for different starting numbers:
Let be the probability for runner to win a race with runners and be the expected starting number of the winner in that race. It can be shown that, for example,, , and for a race with runners.
You are given that , and .
You are given that , and .
Find rounded to 4 digits after the decimal point.