# Transitive and probability

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

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

$C$ is brother of $A$ with probability of $P_3$;

then,

How we can find $P_3$ based on $P_1$ and $P_2$ ?

UPDATE: All events are independent

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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. –  Michael Greinecker Oct 17 '12 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 $A$). –  Hagen von Eitzen Oct 17 '12 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.