Random variables of Uniform distribution

Question

Let $X$ and $Y$ be two independent random variable such that $X$ ~ $U(0,2) $ and $Y$~ $U(1,3)$ then $P(X<Y)$ equals

how do we approach this question..

Answer 1

It is easier to compute $P(X \geq Y)$ first. This probability is $\int_1^{2}\int_y^{2}\frac 1 4dxdy=\frac 18$. Hence $P(X<Y)=1-\frac 1 8=\frac 7 8$

[Note that $0<x<2,1<y<3$ and $y \leq x$ can be rewritten as $y\leq x<2, 1<y<2$]