Law of Total Probability for Conditional Probability given two or more events

Question

I have a question about using the law of total probability for the conditional probability that has more than 1 condition. Here's what I mean, is $P(A | B) = \sum_{i=0}^n P(A | B \cap C_i) \cdot P(C_i)$, where $C_i$'s are mutually exclusive and collectively exhaustive? Why? Can someone give an explanation about this? Does anyone have recommendations for good resources to learn conditional probability?

Answer 1

Actually it must be like this:

$P(A \mid B) = \sum_i P(A \mid C_i \cap B) P(C_i\mid B)$

If $C$ and $B$ are independent of each other we can write it as you did.

$P(A \mid B) = \sum_i \frac{P(A \cap C_i \cap B)}{P( C_i \cap B)} \frac{P(C_i\cap B)}{P(B)}= \sum_i \frac{P(A \cap C_i \cap B)}{P(B)}=\frac{P(A \cap B)}{P(B)}= P(A \mid B)$

I suggest the book "A first course on probability" by Sheldon M. Ross.

Also you can check this link: https://ocw.mit.edu/courses/mathematics/18-05-introduction-to-probability-and-statistics-spring-2014/readings/MIT18_05S14_Reading3.pdf