Refer to the Canadian Journal of Civil Engineering (January 2013) study of variable speed limits, Exercise 4.23 (p. 201). Recall that a portion of an urban freeway was divided into three sections of equal length, and variable speed limits were posted (independently) in each section. The probability distribution of the optimal speed limit for each section follows (probabilities in parentheses). Section 1: 30 mph (.05), 40 mph (.25), 50 mph (.25), 60 mph (.45); Section 2: 30 mph (.10), 40 mph (.25), 50 mph (.35), 60 mph (.30); Section 3: 30 mph (.15), 40 mph (.20), 50 mph (.30), 60 mph (.35). A vehicle adhering to the speed limit will travel through the three sections of the freeway at a steady (fixed) speed. Let x represent this speed.

a. List the possible values of x.

b. Find P(x = 30). [The event {x = 30} is the union of the events {x = 30 in Section 1}, {x = 30 in Section 2}, and {x = 30 in Section 3}. Also, P(x = 30 in Section 1) = P(x = 30 | Section 1) * P(Section 1), where P(Section 1) = 1>3, since the sections are of equal length.]

c. Find the probability distribution for x.

d. What is the probability that the vehicle can travel at least 50 mph through all three sections of the freeway?