Pick two random variables as follows. First pick $X$ uniformly on (0,1), this gives some observed value $X = x$, then pick $Y$ uniformly on $(0, x)$. What's the probability that $X \geq 1/2$ if $Y = 1/4$?
I know $P(X \geq 1/2 | Y = 1/4) = \frac{P(X \geq 1/2)P(Y = 1/4 | X \geq 1/2)}{P(Y = 1/4)}$. Though $P(X \geq 1/2)$ can be easily obtained, since $Y$ is continuous, probability of $Y$ equal to certain point would just be zero. How to derive $P(Y = 1/4 | X \geq 1/2)$ and $P(Y = 1/4)$?