Small Triangle in the Unit Square

The condition $X+Y<1/2$ describes a region inside the unit square.

Sketch the line $x+y=1/2$; the favorable region is a right triangle.

1. 1. Identify the region

   The event $X+Y<1/2$ corresponds to the triangle with vertices $(0,0)$, $(1/2,0)$, $(0,1/2)$, which lies entirely inside the unit square.

2. 2. Compute the area

   This right triangle has legs of length $1/2$, so its area equals the probability.

   $$P(X+Y<1/2)=\frac{1}{2}{\left(\frac{1}{2}\right)}^2=\frac{1}{8}$$

Answer: 1/8