Conditional Region in the Unit Square

Conditioned on $X+Y<1$, the point $(X,Y)$ is uniform on the triangle

which has area $\tfrac12$. The conditional probability is therefore the area of the part of $T$ satisfying $x-y>\tfrac12$, divided by $\tfrac12$.

That part of $T$ is a triangle with the following sides:

• it lies above $y=0$;
• it lies below the line $y=x-\tfrac12$;
• it lies below the line $x+y=1$.

The two sloping lines meet where $x-\tfrac12=1-x$. This gives the point $(\tfrac34,\tfrac14)$.

So the triangle has vertices $(\tfrac12,0)$, $(1,0)$ and $(\tfrac34,\tfrac14)$. Its base is $\tfrac12$ and its height is $\tfrac14$, so its area is

Hence