I'm having some issues solving this question, hopefully someone can help out.
There are 10 girls and 5 boys in a classroom. Each of the 15 kids plays against the others exactly one time, where in each game there's only one winner. Each participant has a 0.5 probability to win. X is the total number of wins of the boys. What is the minimal value of Var(X)? (Note there may be a dependence between the games' results)
I tried finding the expected value which wasn't too difficult, since there's a total of 10 games the boys play among themselves, which guarantees 10 wins for the boys, and each of the boys plays against 10 girls - solving this is as easy as summing the expected values of 5 binomial distributions.
Finding E[X^2] wasn't as easy, so I tried using the formula for the sum of correlated variables, but I don't know the covariances of the Xi,Xjs.
Would love some help here. Thanks!
The minimal variance is zero. Place the children in a circle. Flip a coin. If it shows heads, everyone beats all players to their right. If it shows tails, everyone beats all players to their left. Thus, every game has a $50\%$ chance of going either way, and all participants are certain to win exactly $7$ games.