Ranking Guessing Game

[Guest]

You will independently sample 10 times uniformly at random from the interval [0, 1]. For example, 0.767, 1/π, and 0.25 are all possible values you could find. At the end of this procedure, we will order the samples from least to greatest. Before sampling any numbers, I offer you the following proposition – I’ll pay you \$100 if, after seeing only the first sample (call it x), you can guess what position x will eventually fall in the rankings. For example, if you guess that x will be 3rd smallest, and after seeing the rest of the numbers, this turns out to be true, then I’ll pay you \$100. If I charge you to participate in this game, what is the maximum amount of money you would be willing to pay in exact dollars and cents?