Suppose you have a tournament consisting of 100 players in which each player plays each other player exactly once. In any given match, the chance of either playing winning is equally likely, and there are no ties. Winning a match gives $1$ point, and losing a match gives no points.
After all matches have been played, we rank the players according to their score (from highest to lowest), with ties broken at random. For each $n=1,2,\dots, 100$, let $S_n$ be the score of the player ranked $n$ out of $100$.
Question: What is the expected value of $S_1$? If it's difficulty to find this analytically, how can we approximate the value? The same question holds for other variables, e.g. $S_{10}$, $S_{50}$.
Each individual player's score follows a binomial distribution, but these aren't independent: for instance, the total number of points is fixed. One can bound $S_n$ through a bit of combinatorics: e.g., given that there are $4950$ points in total, we must have $S_1 \geq 50$, and we can construct a scenario in which $S_1 = 99$ (which is maximal), so perhaps $S_1 \approx 75$, the midpoint? If I'm looking at $S_{10}$ instead, similar considerations give what I think is $45 \leq S_{10} \leq 94$. But I'm not sure how to evaluate $\mathbb E(S_{10})$ without further considerations.