A bad player hits a dartboard represented by the unit circle with uniform probability over its area, and a good player has uniform distance distribution over [0,1].
But what's the difference between the two, if every point in the circle is some measure of distance away based on the radius?
Best Answer
The probability that the bad player lands in some region of the dart board is equal to the area of the region divided by $\pi$. So the probability that the bad player lands in the circle of radius $r$ (where $r < 1$) is $\frac{\pi r^2}{\pi} = r^2$. The probability that the good player lands in the circle of radius $r$ is $r$, since her distance to the center is uniformly distributed on $[0,1]$.