Question

In the previous problem, we assumed the distance of the shot from the center was uniformly distributed. As an alternative to the previous problem, now suppose that the shot itself is uniformly distributed in a disk of radius R centered on the target.

The previous problem states: "Consider a person shooting at a target. The score for a given shot will depend on the distance from the target that it hits. We award a score of 10 points if the shot hits within 1 inch of the target, 6 if it hits between 1 and 3 inches of the target, 2 points if it hits between 3 and 5 inches of the target, and 0 points otherwise. Suppose that the distance of the shot from the target is uniformly distributed between 0 and D inches."

(a) In this case, what is the expected value for the score if the person is now also shooting three darts?

(b) What must R be for the expected score, after shooting three darts, to be at least 15?

Answer 1

a) Here the probabilities are proportional to the area.

The probability the shot hits within 1 inch of the target is   

The probability the shot hits between 1 and 3 inches of the target is   

The probability the shot hits between 3 and 5 inches of the target is   

The expected value of the score in single shot is

The expected value for the score if the person is shooting three darts is

b) WE need the expected score to be at least 15. That is

The value of must be at most .