reto wrote:

A game at the state fair has a circular target with a radius of 10 cm on a square board measuring 30 cm on a side. Players win prizes if they throw two darts and hit only the circular area on at least one of the two attempts. What is the probability that Jim won the game?

A. 1 - \(\frac{9-\pi}{9}\)

B. 1- \(\frac{18\pi-\pi^2}{81}\)

C. 9-\(\frac{\pi}{9}\)

D. \(\frac{(9-\pi)^2}{81)}\)

E. \(\frac{18\pi-\pi^2}{81}\)

Fun question. Too bad I missed it on the first attempt.

To solve:

p(success) = 1 - p(failure), where p(failure) = 1 - (Area of Target/Area of Square)

p(success) = \(1 - (1-\frac{10^{2}\pi}{30^{2}})^{2}\).

\(1-(1-\frac{\pi}{9})^{2}\)

\(1-(\frac{9-\pi}{9})^{2}\)

\(1-(\frac{81-18\pi+\pi^{2}}{81})\)

\(\frac{81-81+18\pi-\pi^{2}}{81}\)

\(\frac{18\pi-\pi^{2}}{81}\)

Answer E