The two windscreen wipers on a car windscreen cover respectively 1 2

and 1 3

of the total windscreen area. The overlap area that is covered by both wipers is 1 12 of the total area. A raindrop falls anywhere on the screen with equal probability. Define events A and B for whether the raindrop hits the windscreen at a point covered by the first and second wipers, respectively.

a. What sample space are we implicitly assuming for this question?

b. Write down the probabilities of A and B, and of any combinations of A and B that you can deduce from the information given in the question.

c. Are events A and B independent? Show your working.

d. Give a sentence to explain why you would expect the answer you got for part (c).

e. Find the conditional probabilities, P(A | B) and P(B | A).

For each, compare your answer with the appropriate choice of unconditional probability, either P(A) or P(B). Say whether the conditional probability is larger or smaller than the unconditional probability, and why you would expect this to happen.