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Problem 27

AMC 10/12, early questions

Geometry Difficulty 3.1 Find the answer

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?
$\textbf{(A)}\ \frac{1}{36} \qquad \textbf{(B)}\ \frac{1}{12} \qquad \textbf{(C)}\ \frac{1}{6} \qquad \textbf{(D)}\ \frac{1}{4} \qquad \textbf{(E)}\ \frac{5}{18}$

Your answer

Official 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: $\frac{1}{36} + \frac{2}{36} = \frac{3}{36} = \frac{1}{12} \rightarrow \boxed{\textbf{B}}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.