Customers who purchase a certain make of car can order an engine in any of three sizes. Of all cars sold, 45% have the smallest engine, 35% have the medium-sized one, and 20% have the largest. Of cars with the smallest engine, 10% fail and emissions test within two years of purchase, while 12% of those with the medium size and 15% of those with the largest engine fail. Required:

a. What is the probability that a randomly chosen vehicle will fail an emission test within two years?

b. If a vehicle chosen at random fails an emission test, what is the probability that it has a small engine?

c. If you choose 20 vehicles at random from the total population of vehicles, what is the probability that you choose at most 2 with the largest engine?

d. If you choose 5 vehicles at random from the total population of vehicles every day for 4 days, what is the probability that you choose one vehicle exactly each day with the largest engine?