Probability Problems
Problem Sheet-III
1. If X is uniformly distributed over (0, 10), calculate the probability that a. X < 3 (Ans: 3/10) b. X > 6 (Ans: 4/10) c. 3 < X < 8. (Ans: 5/10)
2. Buses arrive at a specified stop at 15-minute intervals starting at 7 AM. That is, they arrive at 7, 7:15, 7:30, 7:45, and so on. If a passenger arrives at the stop at a time that is uniformly distributed between 7 and 7:30, find the probability that he waits d. Less than 5 minutes for a bus e. More than 10 minutes for a bus.
3. A recent study indicates that the annual cost of maintaining and repairing a car in a town in Ontario averages 200 with a variance of 260. If a tax of 20% is introduced on all items associated with the maintenance and repair of cars (i.e., everything is made 20% more expensive), what will be the variance of the annual cost of maintaining and repairing a car?
(Ans: 374)
4. The time to failure of a component in an electronic device has an exponential distribution with a median of four hours. Calculate the probability that the component will work without failing for at least five hours. (Ans: 0.42)
5. A company has two electric generators. The time until failure for each generator follows an exponential distribution with mean 10. The company will begin using the second generator immediately after the first one fails. What is the variance of the total time that the generators produce electricity? (Ans: 200)
6. In a small metropolitan area, annual losses due to storm, fire, and theft are assumed to be independent, exponentially distributed random variables with respective means 1.0, 1.5, and 2.4. Determine the probability that the maximum of these losses exceeds 3. (Ans: 0.414)
7. The number of days that elapse between the beginning of a calendar year and the moment a high-risk driver is involved in an accident is exponentially distributed. An insurance company
1. If X is uniformly distributed over (0, 10), calculate the probability that a. X < 3 (Ans: 3/10) b. X > 6 (Ans: 4/10) c. 3 < X < 8. (Ans: 5/10)
2. Buses arrive at a specified stop at 15-minute intervals starting at 7 AM. That is, they arrive at 7, 7:15, 7:30, 7:45, and so on. If a passenger arrives at the stop at a time that is uniformly distributed between 7 and 7:30, find the probability that he waits d. Less than 5 minutes for a bus e. More than 10 minutes for a bus.
3. A recent study indicates that the annual cost of maintaining and repairing a car in a town in Ontario averages 200 with a variance of 260. If a tax of 20% is introduced on all items associated with the maintenance and repair of cars (i.e., everything is made 20% more expensive), what will be the variance of the annual cost of maintaining and repairing a car?
(Ans: 374)
4. The time to failure of a component in an electronic device has an exponential distribution with a median of four hours. Calculate the probability that the component will work without failing for at least five hours. (Ans: 0.42)
5. A company has two electric generators. The time until failure for each generator follows an exponential distribution with mean 10. The company will begin using the second generator immediately after the first one fails. What is the variance of the total time that the generators produce electricity? (Ans: 200)
6. In a small metropolitan area, annual losses due to storm, fire, and theft are assumed to be independent, exponentially distributed random variables with respective means 1.0, 1.5, and 2.4. Determine the probability that the maximum of these losses exceeds 3. (Ans: 0.414)
7. The number of days that elapse between the beginning of a calendar year and the moment a high-risk driver is involved in an accident is exponentially distributed. An insurance company