First we calculate for the area of the triangle:
A = s^2 * sqrt(3) / 4
A = 3^2 * sqrt(3) / 4
A = 3.9
Then we calculate the area of a sector where from 1 side
to the other side where distance or r = 1.
Asector = (θ/360) π r^2
where theta is angle of corner a. In equilateral
triangles, all angles are equal and is equivalent to θ = 60°, therefore:
Asector = (60/360) π (1)^2
Asector = 0.52
The probability that x is no more than 1 unit from vertex
a is therefore:
P = Asector / A
P = 0.52 / 3.9
P = 0.13 = 13%
