TITLE: Probability, conditional on a zero probability event
QUESTION [20 upvotes]: Is there a way to resolve probability of an event, given another event that never happens? Mathematically speaking the problem is:
Given that $P(B) = 0$,
$$P(A|B)=\frac{P(A \cap B)}{P(B)} = \frac{0}{0}$$
Is this probability vacuously $0$ of $1$? Can we show that it's one or the other?

REPLY [10 votes]: The definition of conditional probability
$$P(A|B)=\frac{P(A \cap B)}{P(B)}$$
means that we can't condition on an event with zero probability; if $P(B)=0$, then $P(A|B)$ is undefined for any event $A$.
Sometimes it makes sense to define conditional probability differently. Let $X$ and $Y$ be jointly continuous random variables; then we can define conditional PDF of $X$ given $Y$
$$f_{X|Y}(x|y)=\frac{f_{X,Y}(x,y)}{f_Y(y)}$$
for all $y$ such that $f_Y(y)>0$, and using this definition we can define
$$P(X\in A|Y=y)=\int_Af_{X|Y}(x|y)dx$$
which implies conditioning on zero probability event $\{Y=y\}$.
The answer to the general OP question is: if $P(B)=0$, the conditional probabilities given event $B$ are undefined, unless a different definition of conditional probability is used.