1. QBM117 Business Statistics
Probability and Probability Distributions
Revision 1

2. Objectives
To recognise the correct technique for solving probability problems.
To learn how to use the probability flow chart to assist in determining the appropriate distribution to use when solving problems. 2

3. Question 1
An important part of customer service responsibilities of a telephone company relates to the speed with which problems in residential service can be repaired. Suppose past data indicate that the likelihood is 0.70 that a problem in residential service can be repaired on the same day.
For the first five problems reported on a given day, what is the probability that
all five will be repaired on the same day?
at least 3 will be repaired on the same day?
none will be repaired on the same day?
For the first five problems reported on a given day what is the expected number of problems that will be repaired on the same day?

4. Question 2
Battery manufacturers compete on the basis of the amount of time their product lasts in cameras and toys. A manufacturer of alkaline batteries has observed that its batteries last for an average of 26 hours when used in a toy racing car. The amount of time is normally distributed with a standard deviation of 2.5 hours.
What is the probability that a battery lasts
between 24 hours and 28 hours?
longer than 24 hours?
less than 20 hours?
What length of time will at least 90% of the batteries last?

5. Question 3
A survey of top executives revealed that 35% of them regularly read Time magazine, 20% read Newsweek, and 10% of them read both Time and Newsweek.
What is the probability that a particular top executive reads either Time or Newsweek?
Given than a top executive reads Newsweek, what is the probability that they read Time?
Are the events mutually exclusive? Explain.
Are the events independent? Explain.
What is the probability that a particular top executive reads neither Time or Newsweek?

6. Question 4
The flight time of an airplane traveling Chicago to New York is normally distributed from 120 minutes to 140 minutes.
What is the probability of a flight time of less than 125 minutes?
What is the probability of a flight time between 128 and 136 minutes?
What is the expected flight time?

7. Question 5
Airline passengers arrive randomly and independently at the passenger screening facility at Sydney International Airport. The mean arrival rate is 10 passengers per minute.
What is the probability that there are
no arrivals in a 1 minute period?
three of fewer arrivals in a 1 minute period?
no arrivals in a 15 second period?
at least one arrival in a 15 second period?

8. Question 6
The waiting time at a certain bank is normally distributed with a mean of 3.7 minutes and a standard deviation pf 1.4 minutes.
What is the probability that a customer has to wait no more than 2 minutes?
What is the probability that a customer has to wait between 4 and 5 minutes?
20% of customers will have to wait longer than how many minutes?

9. Question 7
A customer service supervisor regularly conducts a survey of customer satisfaction. The results of their last survey indicate that 8% of customers were not satisfied with the service they received at their last visit to the store. Of those who are not satisfied, only 22% return to the store within the year. Of those who are satisfied, 64% return within the year. A customer has just entered the store. He informs you that it is less than 1 year since his last visit to the store. What is the probability that he was satisfied with the service he received on his last visit to the store?

10. Question 8
Large sheets of plastic are cut into smaller pieces to be pressed into credit cards. One manufacturer uses sheets of plastic known to have approximately 3 defects per square meter. The defects occur randomly and independently
An inspector examines a randomly chosen square meter. What is the probability of the inspector finding
no defects?
more that 2 defects?
at most 3 defects?
What is the expected number of defects per square meter?

11. Question 9
5% of a batch of invoices being audited contain errors. A random sample of 10 invoices is taken from the batch.
What is the probability that the sample contains the following:
two or less invoices with errors?
less that two invoices with errors?
two or more invoices with errors?
more than two invoices with errors?
exactly two invoices with errors?
What is the expected number of invoices with errors?

12. Question 10
Psychologists believe that there is a relationship between aggressiveness and order of birth. To test this belief, a psychologist chose 500 primary school students at random and administered each student to a test designed to measure aggressiveness. Each student was classified into one of four categories. The results are shown in the table below.
First born
Not first born
Aggressive 75
Not Aggressive
125
225

13. If a student is chosen at random from the 500
what is the probability that the student is first born?
what is the probability that the student is aggressive?
what is the probability that the student is aggressive given that the student was first born?
Is the event that a student is aggressive independent of the event that the student is first born?

14. Reading for next lecture
Chapters 6 and 7 14