Problem 102 Solution

The density function of the distance to the center is the circumferance of the circle, centered about the center of the dart board, that the dart hits divided by the area of the dart board. Let x be the distance to the center and f(x) the density function.

f(x)=2*pi*x/pi = 2x.

The mean distance to the center is the integral from 0 to 1 of f(x)*x:

Integral from 0 to 1 of 2x*x = 2x³/3 from 0 to 1 = 2/3.

Michael Shackleford, A.S.A.