When an automobile is stopped by a roving safety patrol, each tire is checked for tire wear, and each headlight is checked to see whether it is properly aimed. Let X denote the number of headlights that need adjustment, and let Y denote the number of defective tires.

a. If X and Y are independent with p,(0) = .5. px(1) = .3, P(2) = 2, and p,(0) = .6, py(1) = .1,p,(2) = p,(3) = .05, and py(4) .2, display the joint pmf of (X, Y) in a joint probability table. 0123 0 1 2 3 .08 .07 .04 .00 .06 .15 .05 .04 .05 .04 .10 .06 .00 .03 .04 .07 .00 .01 .05 .06

a. What is P(X1, X21), that is, the probability that there is exactly one customer in each line?

b. What is P(X = X2), that is, the probability that the numbers of customers in the two lines are identical?

c. Let A denote the event that there are at least two more cus- tomers in one line than in the other line. Express A in terms of X, and X2, and calculate the probability of this event.

d. What is the probability that the total number of customers in the two lines is exactly four? At least four?