Calculate conditional probability by conditioning on another event

Question

I would like to calculate the conditional probability P(A|B).

Can I calculate it base on conditioning on another set of mutually exclusive events {C1, C2, ..., Cn) ?

i.e. something like: P(A|B) = P(A|C1)*P(C1|B) + P(A|C2)*P(C2|B) + ... + P(A|Cn)*P(Cn|B)

Thanks for help!

Answer 1

Yes, you can. The formula should be $$P(A | B) = P((A \cap \Omega) | B) = P((A \cap(\cup_n C_n))|B) = P(\cup_n(A \cap C_n))|B)\\ =\sum_nP(A\cap C_n|B) = \sum_nP(A | C_n,B)P(C_n|B)\\ = P(A | B, C_1)P(C_1|B) + P(A | B, C_2)P(C_2|B) + \cdots P(A | B, C_n)P(C_n|B). $$