Question: A chess tournament has 100 players. In the first round, they are randomly paired to determine who plays whom (so 50 games are played). In the second round, they are again randomly paired, independently of the first round. In both rounds, all possible pairings are equally likely. Let X be the number of people who play against the same opponent twice.

(a) Find the expected value of X.

(b) Explain why X is not approximately Poisson.

(c) Find good approximations to P(X = 0) and P(X = 2), by thinking about games in the second round such that the same pair played each other in the first round.