Conditional probability with the condition expressed as a partition

Question

Suppose I want to calculate the following probability:

$$ P(A|B) $$

And I know a partition of $B$, that is $D_1$, $D_2$, $D_3$, ...

How can I express the probability $P(A|B)$ in terms of the probabilities of the partition of $B$: $P(D_1)$, $P(D_2)$ ...

Answer 1

Following @hardmath comment:

$$ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\sum P(A \cap D_i)}{\sum P(D_i)} $$

Answer 2

Hint: if $G$ is an event, then $P(G)$ $ =$ $\sum _ { i } P(G \cap D_{i} ) $ Use this fact and definition of conditional probability.