Let x represent the number of underinflated tires on an automobile.

(a) Which of the following p(x) functions specifies a legitimate distribution for x, and why are the other two not legitimate?

(i) p(0) = 0.3, p(1) = 0.2,  p(2) = 0.1, p(3) = 0.05, p(4) = 0.05

(ii) p(0) = 0.4, p(1) = p(2) = p(3) = 0.1, p(4) = 0.3

(iii) p(x) = 0.2(3-x) for x = 0, 1, 2, 3, 4

(b) For the legitimate distribution of part (a), determine the long-run proportion of cars having at most two underinflated tires, the proportion of cars having fewer than two underinflated tires, and the proportion having at least one underinflated tire.