# What is the probability of the intersection of two events if one of the events has probability zero?

Given two events $A$ and $B$, where the probability of $B$ is zero, i.e., $\Pr(B) = 0$, what is the probability of the intersection of $A$ and $B$, i.e., $\Pr(A \cap B)$?

Is it undefined? Is it zero?

My reasoning is the following:

Since

$$\Pr(A \mid B) = \frac{\Pr(A \cap B)}{\Pr(B)},$$

we have that $\Pr(A \cap B) = \Pr(A \mid B) \cdot \Pr(B)$. But if $\Pr(B) = 0$, then $\Pr(A \mid B)$ would be undefined. So apparently this means that $\Pr(A \cap B)$ is also undefined. On the other hand, this is contrary to intuition: If an event $B$ has probability zero, then it seems to me that the probability of this event occurring AND some other event $A$ occurring should also be zero, i.e. $\Pr(A \cap B) = 0$.

• $A \cap B \subseteq B$, so $0 \le \mathrm{Pr}(A \cap B) \le \mathrm{Pr}(B) = 0$... Deduce the probability of the intersection. Probability is "defined" for every event. The formula for conditional probability holds only when the denominator is not zero. – GEdgar Feb 2 '15 at 18:31
• @GEdgar probability is defined for measurable events. – Matt Samuel Feb 2 '15 at 19:05