Chapter 6, Problem 6.12STPE

The accompanying dartboard is a square whose sides are of length 6:

The three circles are all centered at the center of the board and are of radii 1, 2, and 3, respectively. Darts landing within the circle of radius I score 30 points, those landing outside this circle, but within the circle of radius 2, are worth 20 points, and those landing outside the circle of radius 2, but within the circle of radius 3, are worth 10 points. Darts that do not land within the circle of radius 3 do not score any points. Assuming that each dart that you throw will, independently of what occurred on your previous throws, land on a point uniformly distributed in the square, find the probabilities of the accompanying events:

a. You score 20 on a throw of the dart.

b. You score at least 20 on a throw of the dart.

c. You score 0 on a throw of the dart.

d. The expected value of your score on a throw of the dart.

e. Both of your first two throws score at least 10.

f. Your total score after two throws is 30.