Lottery game variation - Amount you would pay

Question

Question - You have a lottery ticket with 10 slots. Behind each slot there's a equally distributed number between 0 and 1. Your payout is the largest difference between any of these slots. How much are you prepared to pay for the lottery ticket?

This is an interview question that someone was asked.

I understand that since its uniformly distributed between 0 and 1, the expectation of the value in the slot would be 0.5 but I'm unsure of how we find the expected difference between the slots.

Answer 1

The largest difference is the difference between the maximum and the minimum.

The $10$ numbers in the slots divide $[0,1]$ into $11$ intervals. These all have the same expected length. (To see this, uniformly distribute $11$ points on a circle, then uniformly pick one of them at which to split the circle into an interval.) Thus, the expected lengths of the intervals below the minimum and above the maximum are $\frac1{11}$. The expected sum of the lengths of the $9$ intervals between the minimum and the maximum is $\frac9{11}$.

Answer 2

Can also derive expected value for the minimum between n Uniform random variables, which comes out to 1/(n+1) and also the expected value for the maximum of n uniform random variables which comes out to n/(n+1).

The difference is -> (n-1)/(n+1), plug n = 10 and we get 9/11.