This is a problem I came up with after seeing a similar game on tiktok.
Suppose you are given a stream of k numbers where each number is sampled from a uniform distribution between 0 and 1. Before you can see the next number in the stream, you have to guess where the current number will rank (1st to kth place) in the final ordered list of all k generated random numbers. What is the probability, in terms of k, of generating a correct ordering?
I'm pretty sure if k=2, the probability is 0.75. Beyond that I have no idea how to continue. Any ideas?