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$A$ 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 $P_3$;


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
up vote 2 down vote accepted

There's not enough information to solve the problem. In particular, it's possible that A and C are brothers but B is not their brother, but we have no information on what the probability of that event might be.

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.

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