A point is chosen at random inside a circle of radius $R$. What is the probability that the distance of the point from the center is less than the distance from the circumference?

Solution

1. Let $r$ be the distance from the center. The distance from the circumference is $R-r$.
2. The condition $r<R-r$ implies $2r < R$, or $r<R/2$.
3. The area of the region satisfying this is $\pi (R/2{)}^2=\pi R^2/4$.
4. The probability is the ratio of this area to the total area $\pi R^2$, which is $1/4$.
5. Hence the answer is (A).