Given X~unif(a, b) and Y~unif(c, d) with a < c < b < d.
What's the probability that Y>X and Y being realized in the interval (c, b)?
Given X~unif(a, b) and Y~unif(c, d) with a < c < b < d.
What's the probability that Y>X and Y being realized in the interval (c, b)?
Here we assume that $X$ and $Y$ are independent random variables with uniform distributions as specified by OP.
Write out the definition of the probability: $ \mathsf{Pr}\left(Y>X, c
Here is one solution without explicit integration. Let $A$ be the event that $Y>X$ and $B$ the event that $c
Next we get a handle on $\mathrm{Pr}(A|B)$. Split the event $A$ into two disjoint events: $A_1$ and $A_2$ where $A_1$ is the event that $Y>X$ and $X
Finally, using $\mathrm{Pr}(A \cap B)=\mathrm{Pr}(A|B)\mathrm{Pr}(B)$ and doing some algebra we find that $\mathrm{Pr}(A \cap B)=\frac{-2ab+b^2+2ac-c^2}{2(a-b)(c-d)}$.