Probability and Statistics Practice Questions

Question 1: Gas Station Probability Distribution

Use the following for questions 1?3:

The probabilities that a gas station will pump gas into 0, 1, 2, 3, 4, or 5 or more cars during a certain 30?minute period are 0.03, 0.18, 0.24, 0.28, 0.10, and 0.17, respectively.

1. (4 points) Do the set of probabilities constitute a pmf? Justify your answer.
2. (4 points) Write and plot the cdf of the number of cars served during the 30?minute period.
3. (4 points) What is the standard deviation of the number of cars served in a 30?minute period at the gas station?  

A survey of customers purchasing statistical software indicated that 10% of customers were dissatisfied. Half of those dissatisfied customers purchased software from vendor A. It is also known 20% of the surveyed customers purchased from vendor A.

4. (4 points) Given the software was purchased from vendor A, what is the probability that a particular customer was dissatisfied?
5. (4 points) What is the probability a customer did not purchase from vendor A and was satisfied?

A stockbroker believes that under present economic conditions a customer will invest in tax? free bonds with a probability of 0.6, will invest in mutual funds with a probability of 0.3, and will invest in both tax?free bonds and mutual funds with a probability of 0.15.  

6. (4 points) Find the probability the customer will invest in either tax?free bonds or mutual funds.

7. (4 points) Find the probability the customer will invest in neither tax?free bonds or mutual funds.
   
A telephone company operates three identical central office stations. During a one year period, the number of malfunctions reported by each station and the causes are shown below.

8. (4 points) If a malfunction is reported, what is the probability it occurred at Station A and was due to computer malfunction?

9. (4 points) If a malfunction is reported, what is the probability it occurred at Station C?  
10. (4 points) Suppose a malfunction was reported and it was found to be caused by other human errors. What is the probability is came from Station B?

In a certain assembly plant, three machines A, B, and C make 30%, 45%, and 25% of the products, respectively. It is known from past experience the 2%, 3%, and 2% of the products made by each machine, respectively, are defective.

11. (4 points) If a finished product is selected at random, what is the probability it is defective?
12. (4 points) What is the probability a randomly selected product is made on machine A and is also defective?
13. (4 points) What is the probability that, if a defective product is selected, it was made on machine C?

In testing a certain kind of truck tire over rugged terrain, it is found that 25% of the trucks fail to complete the test run without a blowout. Answer the following questions related to the next 15 trucks that are tested.

14. (4 points) What is the probability that fewer than 3 but at least one truck will have a blowout?
15. (4 points) What is the expected value for the number of trucks with blowouts?
16. (4 points) Suppose exactly 13 trucks have blowouts. Do you feel that the 25% figure stated above is correct? Why or why not?
17. (4 points) Suppose that airplane engines operate independently and fail with probability equal to 0.4. Assuming that a plane makes a safe flight if at least one?half of its engines run, determine whether a 4?engine plane or a 2?engine plane has the higher probability for a successful flight.

An electrical circuit is detailed in the figure below. The probability each component works is provided in the associated blocks in the diagram. Assume the components operate independently.

18. (4 points) What is the probability that the entire system works?
19. (4 points) Given that the system works, what is the probability that component A is not working?
20. (4 points) The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 of the next 7 patients having this operation survive?
21. (4 points) A private pilot wishes to insure his airplane for$200,000. The insurance company estimates that a total loss will occur with probability 0.002, a 50% loss with probability 0.01, and a 25% loss with probability 0.1. Ignoring all other partial losses, what premium should the insurance company charge each year to realize an average profit of $500?

22. (4 points) A coin is biased such that a head is three times as likely to occur as a tail. Find the expected number of tails when this coin is tossed twice.

We conduct an experiment in which two fair dice are rolled (note: each die can land on 1, 2, 3, 4, 5, or 6).

23. (4 points) Write the sample space for the experiment

24. (4 points) Let the random variable X be the sum of the two numbers on each die. Write the pmf for X.

25. (4 points) What is the expected value of X?