Here's the problem I'm stuck on:
There are d people competing in a dance competition. Each dancer will compete in a dance-off with everyone else in the competition. They are all equally skilled and have an equal chance of winning any matchup. What is the probability that dancer 1 will win exactly half of their dance matchups? You must prove your answer, do not merely give examples. Assume d is odd.
So, I've figured out that half of the dance matchups for a dancer will be (d-1)/2. I think there should be a total of d(d-1)/2 matchups, and each dance off has a 1/2 probability of being either a win or a loss. But I also think each dancer's number of wins is not independent of the other's numbers of wins, so I am confused on how I can find out the probability that a person will win exactly half of their dances by combining my information somehow.