for a project I want to compare the observed distribution of wins of sport teams with an all luck scenario where all teams in a league have a 50/50 chance of winning each game.
More precisely the observed distribution can be estimated from the results. For example if we have 4 Teams and everybody is playing each other we have 6 Games and the results could look like this:
| Team | #Wins |
|---|---|
| Team A | 3 |
| Team B | 1 |
| Team C | 1 |
| Team D | 1 |
If Teams are differently skillfull the more games you play the more unlikely it is that this is not reflected in the results. In the scenario where every Team has a 50/50 chance of winning each game if we play an infinite amount of games all teams will approximately have the same number of wins. However if we play just $n$ games we can expect some variance. Is there a way to quantify the expected variance?
I was thinking about using the binomial distribution to estimate the number of wins per team. If I pick one Team and want to estimate the number of wins in $n$ games $B(n,0.5)$ seems like a reasonable choice. However If I want to do that for multiple teams I have the problem that the distributions are not independent. More precisely:
Let $x$ be the vector with the number of wins of each team $x=[W_A, W_B, W_C, W_D]$ $$x = [B(n,0.5),B(n,0.5),B(n,0.5),B(n,0.5)]$$
assumes the events to be independent. However I have to account for the fact that the number of total wins is always $\sum_{k=1}^{n-1}(n-k) = \frac{1}{2}n(n-1)$.
Is there a way to account for that condition in my calculations?
If yes, can this be extended to scenarios where Teams not necessarily play the same amount of games or certain teams play each other twice while other teams are not playing against each other not at all as it is the case in some north american sports leagues?
Edit: Just to clarify: I am looking for the variance of the wins of all teams $x$ not only of one Team $W_A$.