Conditional Probability of A given B

Question

Very basic question, but If A and B are independent events, then is the Probability of A given B has occurred = Zero or Probability of A?

and why?

Answer 1

The conditional probability is given by $P(A|B) = \frac{P(A\cap B)}{P(B)}$. And events $A$ and $B$ are independent if $P(A|B) = P(A)$ (in words, the probability that $A$ occurs given that $B$ occurs is that same as $A$ occurs with regardless of knowledge of $B$). Equivalently, $P(A\cap B) = P(A)P(B)$.

So the answer is $P(A|B) = \frac{P(A\cap B)}{P(B)} = P(A)$. Not $P(A)=0.$

If you have events with $P(A\cap B) = 0$ and $P(A \cup B) = 1$, they are called mutually exclusive. In that case you could conclude that $P(A|B) = 0$. But mutually exclusive events are highly dependent.

Mutually exclusive events are in some sense the opposite of independent events. If $A$ and $B$ are mutually exclusive, and you know whether $A$ occurred, then you know for sure whether $B$ occurred. Whereas if $A$ and $B$ are independent events, if you know that $A$ occurred, you have no information about whether $B$ occurred. For independent events, you have $P(A|B)=P(A).$