Let’s consider each side one at a time. Let’s begin with side a. If a is the longest side it means that b+c > a. That means that the two opposite sides together must be longer. If they are not, then a triangle is not formed. When a = b+c it means that the opposite sides just touch and the 3 line segments overlap to form a single line segment. When a > b+c, the ends cannot touch to form a triangle.

For simplicity we used the interval [0,1]. Thus a can be anything from 0 to 1. For any given a we can draw a plot of the line a = b+c. This line separates the two regions of success and failure. The line intercepts the b-axis at a and the c-axis at a. The failure area is a triangular region that runs from (0, 0) to (a,0) and then to (0, a). When a is a maximum, i.e. 1, then the area is 1/2. The failure volume is a 3-sided pyramid with height = 1 and base = 1/2. Thus the volume of the failure region is 1/3Bh = 1/3(1/2)1 = 1/6.

The volume of the entire solution space = 1, the volume of a cube of side 1. The volume of the failure region for a = 1/6. The same is true of the failure region for b and c.

Thus the volume of the success region = 1 – 3(1/6) = 1/2

The probability of success = V(success)/V(all possibilities) = (1/2)/1 = 1/2