Are conditional probabilities only work on uniform distribution?

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The conditional probability of $A$ given $B$ is defined by $\mathrm{Pr}(A\mid B)=\mathrm{Pr}(A\cap B)/\mathrm{Pr}(B)$. I'm trying to understand this intuitively, but the intuition only works for the uniform distribution sample space, so I'm suspicious that there're conditional probabilites for not uniform spaces. Maybe my concepts are very restricted since I'm a beginner in probability theory. If so, what are the examples?

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There's a typo in the formula: the numerator should be the probability of intersection. Intuitively, you are "reducing" your space to $B$ ($B$ is now your whole space, if you condition it), hence the probability of $A$ know $B$ happened is the probability of $A$ (inside $B$, that is the intersection) divided by the probability (or the "size") of your new whole space, which is $B$,