# Transitive and probability

$A$ is brother of $B$ with probability of $P_1$;

$B$ is brother of $C$ with probability of $P_2$;

How we can find probability of $A$ is brother of $C$ ($P_3$), based on $P_1$ and $P_2$?

All events are independent.

• We can't. For example, even under some idependence assumptions, you cannot get the probability that $C$ is a brother of $A$ but $B$ is not. Oct 17, 2012 at 13:54
• No way. We don't now if the first two events are independant. Even if we knew that $A$ is a brother of $B$ and $B$ a brother of $C$ we could not infer that $C$ is a brother of $A$ ($C$ might be a sister of or identical to $A$). Oct 17, 2012 at 13:56

If we assume that this probability is zero (or negligibly small), and if the events "A is a brother of B" and "B is a brother of C" are independent, then the answer is trivial: $P_3 = P_1 P_2$. But that's a lot of unstated assumptions.