Hard

Probability

You generate a uniformly random number in the interval $(0,1)$. You can generate additional random numbers as many times as you want for a fee of $\$0.02$ per generation. This decision can be made with the information of all of the previous values that have been generated. Your payout is the maximum of all the numbers you generate. Under the optimal strategy, find your expected payout of this game.

Notes

Hint