AMC10 2011 10A, Problem 14

A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?

Solution: For the circumference to be greater than the area, we must have $\pi d > \pi \left( \frac{d}{2} \right) ^2$, or $d<4$. Now since $d$ is determined by a sum of two dice, the only possibilities for $d$ are thus $2$ and $3$. In order for two dice to sum to $2$, they most both show a value of $1$. The probability of this happening is $\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$. In order for two dice to sum to $3$, one must show a $1$ and the other must show a $2$. Since this can happen in two ways, the probability of this event occurring is $2 \times \frac{1}{6} \times \frac{1}{6} = \frac{2}{36}$. The sum of these two probabilities now gives the final answer: $\frac{1}{36} + \frac{2}{36} = \frac{3}{36} = \frac{1}{12} \rightarrow \fbox{\frac{1}{12}}$

Answer: \frac{1}{12}