Joint Uniform Distribution Probability Problem Let $X \sim U(0,1) $ and $Y \sim U(0,x) $. Calculate $$ \Pr(X >0.5 | Y= 0.25)$$
Is this a trick question ? Since $\Pr(Y = 0.25) = 0$, right ?

$Y \sim U(0,x)$ is really saying that if $X$ has value $x \in (0,1)$, then the conditional pdf of $Y$ is uniform on $(0,x)$.  That is, for
$0 < y < x < 1$,
$$f_{X,Y}(x,y) = f_{Y\mid X}(y\mid X=x)f_X(x) = \frac 1x \cdot 1 = \frac 1x, 0 < y < x < 1.$$
Note that from this we can find $f_Y(y)$ as
$$f_Y(y) = \int_y^1 f_{X,Y}(x,y) \mathrm dx =  \int_y^1 \frac 1x \mathrm dx = -\ln y, 0 < y < 1.$$
What you really should find is the conditional pdf of $X$ given that $Y=0,25$ from which calculation the desired quantity $P(X>0.5 \mid Y = 0.25)$ (which we recognize as the complementary conditional CDF of $X$ given that $Y = 0.25$) ir readily obtained.  We have that
$$f_{X\mid Y = 0.25}(x\mid Y=0.25) = \dfrac{f_{X,Y}(x,0.25)}{f_Y(0.25)}.$$
Can you finish from here?
