I started reading conditional probability . I found the following equation. I can not understand the equation. $P(A \cap B) = P(A | B) . P(B)$ .Can we prove it for any case? $P(X)$ denotes the probability of happening $X$.
According to the Kolmogrov's definition , $\frac { P(A \cap B)}{P(B)}$ is denoted by the notation $P(A|B)$. If this is so , then how $\frac { P(A \cap B)}{P(B)}$ gives the probability of happening $A$ , when $B$ has happened already.
I do not think it can be proved except for the case when the random experiment is done on a sample space with finite number of points.
I think we have to take this statement as an Axiom. Can anyone please correct me if I am wrong?