Conditional Probability and Division by Zero

This is a surprisingly philosophical question, and as such, here is a link to a philosophical paper about it: What Conditional Probability Could Not Be

Practically speaking though, you're absolutely correct - this probability is $1/2.$ However it is difficult to describe this fact using conditional probability the way it is usually understood.

The way I would "rigorously" approach this problem the following: let's say you have a probability space $(X,\Sigma,P)$ and a subspace $Y\subset X$ such that $P(Y)=0$. How do we 'condition' on this space? Well, the same way we consider a "line" integral in $\mathbb{R}^2$: $Y$ becomes your new universe, so you have to define a new probability space $(Y,\Sigma_2,P_2)$ where you can answer questions such as this. The statement $P(A\cap B)/P(B)$ is somewhat like trying to measure the length of a line segment using a bathroom scale - the scale ignores the line segment, so you have to get a ruler instead!