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my question concerns the law of total probability:

(1) $\;\;P(A) = \sum_{i} P(A|B_i) \cdot P(B_i) \;.$

In a book I found an equation, which seems to be an extension of equation (1) for the case that the conditional probability $P(A|B_i)$ consists of more than one argument on the condition side, like $P(A|B_i, C_j)$. The equation for calculating $P(A)$ for that case is given by

(2) $\;\;P(A) = \sum_{i}\sum_{j} P(A|B_i, C_j) \cdot P(B_j) \cdot P(C_i) \;.$

I guess that (2) holds only true in the case that B and C are stochastically independent?

It would be very nice if someone may help to answer this question, or if somebody knows some literature where something like equation (2) is explained.

Many thanks in advance

user3523846
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1 Answers1

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You are right, the factors on the right-hand side should be $P(B_i,C_j)$. But stochastic independence gives $P(B_i)⋅P(C_j)$.

Wuestenfux
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