conditional probability

$P(A|B)$ is read as “the probability of A, given B”. So assuming B happens, what’s the probability of A?

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

This means, you get the intersection of A and B happening divided by the total probability of B happening.

If events A & B are independent, then $P(A|B)=P(A)$ and $P(B|A)=P(B)$. It makes sense if you think about it, given that independent means one doesn’t influence the other.

Due to the probability of intersection of two indepdendent events, the probability of intersection between two events equals the product of each of them happening. $P(A\cap B)=P(A)P(B)$. Which also means that if that formula works out, then the events are independent.