1. (35 points) You have a bag of 5 dice. One of them have sixes on all six sides. The other four are normal and have one six and one of every other number. You randomly grab two dice out of the bag. (a) What is the probability that one of the two dice is the one with all sixes? ( (c) If you roll two sixes, what is the probability that you have two normal dice? b) If you roll the two dice once, what is the probability you will get two sixes? (d) If you get two sixes on the rst roll, and then roll the two dice again, what is the probability you will get two sixes again? (e) If you get two sixes on the rst roll, what is the probability neither die lands on six on the second roll? (24 points) Five percent of the cars sold by a manufacturer are defective and have a high probability of breaking down quickly. The remaining ninety ve percent tend to last longer. The amount of time (in years) before a defective car breaks down is described by the following probability density function (pdf): pD(t) = 96"!\" The cumulative density function (cdf) is: F905) 2 1 _'9t For normal cars the pdf and cdf are: pN(t) 2 15615: FN(t) = l e_'15t You buy a car from the manufacturer (a) What is the probability your car will last between one and two years? (b) If the car lasts between one and two years, what is the probability it was defective? (c) What is the probability the car lasts more than two years