1. Chapter 4
Created by Bethany Stubbe and Stephan Kogitz

2. A Survey of Probability Concepts
Chapter Four
A Survey of Probability Concepts
GOALS
When you have completed this chapter, you will be able to:
ONE Define probability.
TWO Describe the classical, empirical, and subjective approaches to probability.
THREE Understand the terms: experiment, event, outcome, permutations, and combinations.
FOUR Define the terms: conditional probability and joint probability.

3. A Survey of Probability Concepts
Chapter Four continued
A Survey of Probability Concepts
GOALS
When you have completed this chapter, you will be able to:
FIVE Calculate probabilities applying the rules of addition and the rules of multiplication.
SIX Use a tree diagram to organize and compute probabilities.

4. Definitions It can only assume a value between 0 and 1.
A probability is a measure of the likelihood that an event in the future will happen.
It can only assume a value between 0 and 1.
A value near zero means the event is not likely to happen. A value near one means it is likely.
There are three definitions of probability: classical, empirical, and subjective.

5. Definitions continued
The classical definition applies when there are n equally likely outcomes.
The empirical definition applies when the number of times the event happened in the past is divided by the number of observations.
Subjective probability is based on whatever information is available.

6. Definitions continued
An experiment is a process that leads to the occurrence of one and only one of several possible observations.
An outcome is the particular result of an experiment.
An event is the collection of one or more outcomes of an experiment.

7. Mutually Exclusive Events
Events are mutually exclusive if the occurrence of any one event means that none of the others can occur at the same time.
Events are independent if the occurrence of one event does not affect the occurrence of another.

8. Collectively Exhaustive Events
Events are collectively exhaustive if at least one of the events must occur when an experiment is conducted.

9. Example 1 A fair die is rolled once.
The experiment is rolling the die.
The possible outcomes are the numbers 1, 2, 3, 4, 5, and 6.
An event is the occurrence of an even number. That is, we collect the outcomes 2, 4, and 6.
A fair die is rolled once.

10. EXAMPLE 2
Throughout her teaching career Professor Jones has awarded 186 “A”s out of 1,200 students. What is the probability that a student in her section this semester will receive an A?
This is an example of the empirical definition of probability.
To find the probability a selected student earned the letter grade A:

11. Subjective Probability
Examples of subjective probability are:
Estimating the probability the Calgary Stampeders will play the Grey Cup this year.
Estimating the probability mortgage rates for home loans will top 8 percent.

12. Basic Rules of Probability
If two events A and B are mutually exclusive, the special rule of addition states that the probability of A or B occurring equals the sum of their respective probabilities:
P(A or B) = P(A) + P(B)

13. EXAMPLE 3
Maritime Commuter Airways recently supplied the following information on their commuter flights from Moncton to St. John’s.

14. EXAMPLE 3 continued
If A is the event that a flight arrives early, then P(A) = 100/1000 = .10.
If B is the event that a flight arrives late, then:
P(B) = 75/1000 = .075.
The probability that a flight is either early or late is:
P(A or B) = P(A) + P(B) = =.175.

15. The Complement Rule
The complement rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1.
If P(A) is the probability of event A and P(~A) is the complement of A,
P(A) + P(~A) = 1 or P(A) = 1 - P(~A).

16. The Complement Rule continued
A Venn diagram illustrating the complement rule would appear as: A ~A

17. EXAMPLE 4
Recall EXAMPLE 3. Use the complement rule to find the probability of an early (A) or a late (B) flight
If C is the event that a flight arrives on time, then P(C) = 800/1000 = .8.
If D is the event that a flight is canceled, then P(D) = 25/1000 = .025.

18. EXAMPLE 4 continued P(A or B) = 1 - P(C or D) = 1 - [.8 +.025] =.175 D
~(C or D) = (A or B)
.175

19. The General Rule of Addition
If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula:
P(A or B) = P(A) + P(B) - P(A and B)

20. The General Rule of Addition
The Venn Diagram illustrates this rule: B
A and B A

21. EXAMPLE 5
In a sample of 500 students, 320 said they had a stereo, 175 said they had a TV, and 100 said they had both: TV
175
Both
100
Stereo
320

22. EXAMPLE 5 continued
If a student is selected at random, what is the probability that the student has only a stereo, only a TV, and both a stereo and TV?
P(S) = 320/500 = .64.
P(T) = 175/500 = .35.
P(S and T) = 100/500 = .20.

23. EXAMPLE 5 continued
If a student is selected at random, what is the probability that the student has either a stereo or a TV in his or her room?
P(S or T) = P(S) + P(T) - P(S and T)
= = .79.

24. Joint Probability
A joint probability measures the likelihood that two or more events will happen concurrently.
An example would be the event that a student has both a stereo and TV in his or her residence room.

25. Special Rule of Multiplication
The special rule of multiplication requires that two events A and B are independent.
Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other.
This rule is written: P(A and B) = P(A)P(B)

26. EXAMPLE 6 P(CIBC and Bell) = (.5)(.7) = .35.
Chris owns two stocks, CIBC and Bell Canada. The probability that CIBC stock will increase in value next year is .5 and the probability that Bell stock will increase in value next year is .7. Assume the two stocks are independent. What is the probability that both stocks will increase in value next year?
P(CIBC and Bell) = (.5)(.7) = .35.

27. EXAMPLE 6 continued
What is the probability that at least one of these stocks increase in value during the next year? (This means that either one can increase or both.)
P(at least one) = (.5)(.3) + (.5)(.7) +(.7)(.5)
= .85.

28. Conditional Probability
A conditional probability is the probability of a particular event occurring, given that another event has occurred.
The probability of the event A given that the event B has occurred is written P(A|B).

29. General Multiplication Rule
The general rule of multiplication is used to find the
joint probability that two events will occur.
It states that for two events A and B, the joint probability that both events will happen is found by multiplying the probability that event A will happen by the conditional probability of B given that A has occurred.

30. General Multiplication Rule
The joint probability, P(A and B) is given by the following formula:
P(A and B) = P(A)P(B/A) or P(A and B) = P(B)P(A/B)

31. EXAMPLE 7
The Dean of the School of Business at Owens University collected the following information about undergraduate students in her school:

32. EXAMPLE 7 continued
If a student is selected at random, what is the probability that the student is a female (F) accounting major (A)
P(A and F) = 110/1000.
Given that the student is a female, what is the probability that she is an accounting major?
P(A|F) = P(A and F)/P(F)
= [110/1000]/[400/1000] = .275

33. Some Principles of Counting
The multiplication formula indicates that if there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both.
Example 8: Dr. Delong has 10 shirts and 8 ties. How many shirt and tie outfits does he have?
(10)(8) = 80

34. Some Principles of Counting
A permutation is any arrangement of r objects selected from n possible objects.
Note: The order of arrangement is important in permutations.

35. Some Principles of Counting
A combination is the number of ways to choose r objects from a group of n objects without regard to order.

36. EXAMPLE 9
There are 12 players on the Glacier High School basketball team. Coach Thompson must pick five players among the twelve on the team to comprise the starting lineup. How many different groups are possible?

37. Example 9 continued
Suppose that in addition to selecting the group, he must also rank each of the players in that starting lineup according to their ability.