# Probability for incomplete information

Let's say there are 10 teams: A-J. Only 1 team wins, others lose. Probability of any team to win is unknown (different for each team) and to be calculated. Not all teams participate in each game.

There were 10 games. Teams C-J participated in all of them. Team B participated in all 10 games and won 3 times. Team A participated in 4 games and won 3 times.

What is the probability of team A win?

What is the probability of team B win?

There were 10 games. Teams C-J participated in all of them. Team B participated in all 10 games and won 3 times. Team A participated in 4 games and won 3 times.

What is the probability of team A win?

That depends on whether you want the unconditional probability of winning $(W=A$), or the probability of winning conditional on participation $(W=A\mid A\in {\mathbf G})$.

\begin{align}\Pr(W=A) &= \frac{3}{10} & \text{A won 3 of all 10 games} \\[1ex] \Pr(W=A\mid A\in {\mathbf G}) &= \frac 3 4 & \text{A won 3 of 4 participated games} \end{align}

Notice that the events of other teams playing is at this point a distraction. It is simply a frequentist problem.

What is the probability of team B win?

You can now find: $\Pr(W=B)$

• At 11th game in real life the probability for A is 3/4 but the participation set is smaller, however for B the set is larger but probability is 3/10. I was thinking how to account the fact that B is very new and include "uncertainty" factor into the calculation of the probability. Should I re-phrase my question? – Artem Aug 11 '14 at 7:01