Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate ? = 8 per hour. Suppose that with probability 0.5 an arriving vehicle will have no equipment violations. (a) What is the probability that exactly eight arrive during the hour and all eight have no violations? (Round your answer to four decimal places.)   (b) For any fixed y ≥ 8, what is the probability that y arrive during the hour, of which eight have no violations?       (c) What is the probability that eight "no-violation" cars arrive during the next hour? [Hint: Sum the probabilities in part (b) from y = 8 to ∞.] (Round your answer to three decimal places.)   You may need to use the appropriate table in the Appendix of Tables to answer this question.

In the mid-afternoon a restaurant receives orders such that the time between them follows an exponential distribution. The mean time between orders is 11 minutes. a) What is the probability that the restaurant receives no orders in a 30-minute interval? b) What is the probability that the restaurant receives at least one other order within 1 min of the first one?

23. *An insurance policy pays an individual 100 per day for up to 3 days of hospitalization and 25 per day for each day of hospitalization thereafter. The number of days of hospitalization, X, is a discrete random variable with probability function Pr(X = k) = (1/15)(6 – k), k = 1, 2, 3, 4, 5. Calculate the expected payment for hospitalization under this policy. (213.3)

Q3 a) A component has an exponential time to failure with a mean of 100 hours. (i) The component has already been in operation for its mean life. What is the probability that it will fall by 150 hours? (ii) The component will operate for another 50 hours given that it is in operation 150 hours. (iii) If 20 components are tested, what is the probability that at least one fails in less than 200 hours? Assume that the components fail independently. b) Students of National University were classified according to their favorite color: Blue 12 5 Favorite color Red 13 13 Black 25 20 Male Female If one student is selected at random, then find the probability that the selected student: (i) Doesn't like black color. (ii) Is Female and likes blue color. (iii) Is Male or like black color. (iv) Likes red color given that it is female.