2011 AMC 12A Problem 10

Problem 10 of 25EasierAlgebra Geometry Counting & Probability

A pair of standard

6

-sided fair 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?

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Solution

For diameter

d,

area

\lt

circumference means

\dfrac{\pi d^2}{4} \lt \pi d,

i.e.

d \lt 4.

Since

d \ge 2,

this needs a sum of

2

3.

A sum of

2

has probability

\dfrac{1}{36}

and a sum of

3

has probability

\dfrac{2}{36},

totaling

\dfrac{3}{36} = \dfrac{1}{12}.

Thus, the correct answer is B.

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