Chess Tournament

The Problem

A chess tournament has $n$ players. Each pair plays exactly one game. A win is 1 point, a draw is 1/2 for each, a loss is 0.

1. What is the total number of games played?
2. What is the sum of all players' scores?
3. If all players are equally skilled and each game is independently a draw with probability $p$, win for white with probability $(1-p)/2$, win for black with probability $(1-p)/2$, what is the expected score of a specific player?
4. In a round-robin with $n=8$ where every game ended decisively (no draws), what's the maximum number of players who could be tied for first place?