1
$\begingroup$

Is there anyway, to compute the probability of the event given two dependent events? I know that Bayes can help if those events are independent, but what if condition events are dependent?

$\endgroup$
3
2
$\begingroup$

Let $A$, $B$, $C$ be three (potentially dependent) events. By the probability chain rule: $$ P(A,B,C)=P(A|B,C)P(B,C)=P(A|B,C)P(B|C)P(C) $$ So we can do \begin{align} P(A|B,C) &= \frac{ P(A,B,C)}{P(B|C)P(C)} \\ &= \frac{ P(B,C|A) P(A) }{ P(B|C)P(C) } \\ &= \frac{ P(B|A,C) P(A) P(C) }{ P(B|C)P(C) } \\ &= \frac{ P(B|A,C) P(A) }{ P(B|C) } \end{align}

See this related post and this one as well.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.