What is $\mathbb P(X\in A\mid Y\in B)$?

Question

I'm confuse with formula of conditional probability.

In one hand : $$\mathbb P(X\in A,Y\in B)=\int_B \mathbb P(X\in A\mid Y=y)f_Y(y)dy=\int_B\int_A f_{X\mid Y=y}(x)dxf_Y(y)dy.$$

Now, do we have something as :$$\mathbb P(X\in A\mid Y\in B)=\int_A f_{X|Y\in B}(x)dx \ \ ?$$ or $$\mathbb P(X\in A\mid Y\in B)=\int_B \mathbb P(X\in A\mid Y=y)dy=\int_B\int_Af_{X\mid Y=y}(x)dxdy ?$$

I'm very confuse with all those formulas...

Answer 1

No. What you have is $$\mathbb P(X\in A\mid Y\in B) = \dfrac{\mathbb P(X\in A,Y\in B)}{\mathbb P(Y\in B)} = \dfrac{\int\limits_B\int\limits_A f_{X\mid Y=y}^{\,}(x)f_Y^{\,}(y) \,dx\,dy}{\int\limits_B f_Y^{\,}(y) \,dy}$$ at least when the denominator is non-zero. This cannot be simplified in general