Question

On the dart board shown in the figure below, the outer circle has radius $6$ and the inner circle has radius $3.$ Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values of the regions hit. What is the probability that the score is odd?

$\dfrac{17}{36}$

$\dfrac{35}{72}$

$\dfrac{1}{2}$

$\dfrac{37}{72}$

$\dfrac{19}{36}$

Solution:

The area of the outer circle is $6^2\pi = 36\pi,$ and the area of the inner circle is $3^2\pi = 9\pi.$ Therefore, the area of the outer ring is $36\pi - 9\pi = 27\pi.$

This means that the probability of hitting an outer segment is $\dfrac{9\pi}{36\pi} = \dfrac{1}{4}.$ Similarly, the probability of hitting an inner segment is $\dfrac{3\pi}{36\pi} = \dfrac{1}{12}.$

Summing over all possibilities, the probability of hitting a $1$ is $\dfrac{1}{12} + 2 \cdot \dfrac{1}{4} = \dfrac{7}{12}.$ Similarly, the probability of hitting a $2$ is $2 \cdot \dfrac{1}{12} + \dfrac{1}{4} = \dfrac{5}{12}.$

The only way to get an odd score is to hit one $1$ and one $2.$

The probability of hitting these numbers in either order is $2 \cdot \dfrac{5}{12} \cdot \dfrac{7}{12} = \dfrac{35}{72}.$

Thus, B is the correct answer.

2007 AMC 8 Problem 25

Solution:

Problem 25 in Other Years