Section 2.2 Exercise 4

Suppose you throw a dart at a circular target of radius 10 inches. Assuming that you hit the target and that the coordinates of the outcomes are chosen at random, find the probability that the dart falls

(a) within 2 inches of the center.

\[P(within\ 2\ inches\ of\ center) = \frac{\pi (2)^2}{pi (10)^2} = \frac{4}{100} = 0.04\]

(b) within 2 inches of the rim.

\[P(within\ 2\ inches\ of\ rim) = \frac{\pi (10)^2- \pi (8)^2}{\pi (10)^2} = \frac{36}{100} = 0.36\]

(c) within the first quadrant of the target.

\[P(within\ 1st\ quadrant) = \frac{1}{4} = 0.25\]

(d) within the first quadrant and within 2 inches of the rim.

Using the property \(P(A \cup B) = P(A)P(B)\) for independent events:

\[P(within\ 1st\ quadrant\ and\ within\ 2" of\ rim) = P(within\ 1st\ quadrant\ \cup\ within\ 2\ inches\ of\ rim) = \\ P(within\ 1st\ quadrant) * P(within\ 2\ inches\ of\ rim) = 0.25 * 0.36 = 0.09\]