1. 5- 1 Chapter Five McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.

2. 5- 2 Chapter Five 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. Goals

3. 5- 3 Chapter Five continued A Survey of Probability Concepts GOALS When you have completed this chapter, you will be able to: FOUR Define the terms: conditional probability and joint probability. FIVE Calculate probabilities using the rules of addition and the rules of multiplication. SIX Use a tree diagram to organize and compute probabilities. Goals

4. 5- 4 Movie

5. 5- 5 Movie

6. 5- 6 Definitions continued Event An Event is the collection of one or more outcomes of an experiment. Outcome An Outcome is the particular result of an experiment. Experiment: A fair die is cast. Possible outcomes: The numbers 1, 2, 3, 4, 5, 6 One possible event: The occurrence of an even number. That is, we collect the outcomes 2, 4, and 6.

7. 5- 7 Definitions continued There are three definitions of probability: classical, empirical, and subjective. Classical The Classical definition applies when there are n equally likely outcomes. Empirical The Empirical definition applies when the number of times the event happens is divided by the number of observations. Subjective Subjective probability is based on whatever information is available.

8. 5- 8 Mutually Exclusive Events Mutually Exclusive Events are Mutually Exclusive if the occurrence of any one event means that none of the others can occur at the same time. Mutually exclusive: Rolling a 2 precludes rolling a 1, 3, 4, 5, 6 on the same roll.

9. 5- 9 Collectively Exhaustive Events Collectively Exhaustive Events are Collectively Exhaustive if at least one of the events must occur when an experiment is conducted.

10. 5- 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 an A:

11. 5- 11 Subjective Probability Examples of subjective probability are: estimating the probability the Washington Redskins will win the Super Bowl this year. estimating the probability mortgage rates for home loans will top 8 percent.

12. 5- 12 Basic Rules of Probability P(A or B) = P(A) + P(B) If two events A and B are mutually exclusive, the Special Rule of Addition Special Rule of Addition states that the probability of A or B occurring equals the sum of their respective probabilities.

13. 5- 13 Example 3 New England Commuter Airways recently supplied the following information on their commuter flights from Boston to New York:

14. 5- 14 Example 3 continued The probability that a flight is either early or late is: P(A or B) = P(A) + P(B) =.10 +.075 =.175. 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.

15. 5- 15 The Complement Rule 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). 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.

16. 5- 16 The Complement Rule continued A ~A~A~A~A Venn Diagram A Venn Diagram illustrating the complement rule would appear as:

17. 5- 17 Example 4 If D is the event that a flight is canceled, then P(D) = 25/1000 =.025. 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.

18. 5- 18 Example 4 continued C.8 D.025 ~(C or D) = (A or B).175.175 P(A or B) = 1 - P(C or D) = 1 - [.8 +.025] =.175

19. 5- 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) The General Rule of Addition

20. 5- 20 The General Rule of Addition A and B A B The Venn Diagram illustrates this rule:

21. 5- 21 EXAMPLE 5 Stereo 320 320 Both 100 100 TV175 In a sample of 500 students, 320 said they had a stereo, 175 said they had a TV, and 100 said they had both. 5 said they had neither. Neither 5

22. 5- 22 Example 5 continued P(S or TV) = P(S) + P(TV) - P(S and TV) = 320/500 + 175/500 – 100/500 =.79. P(S and TV) = 100/500 =.20 If a student is selected at random, what is the probability that the student has only a stereo or TV? What is the probability that the student has both a stereo and TV?

23. 5- 23 Joint Probability 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 dorm room.

24. 5- 24 Special Rule of Multiplication   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) Special Rule of Multiplication The Special Rule of Multiplication requires that two events A and B are independent.

25. 5- 25 Example 6 P(IBM and GE) = (.5)(.7) =.35 Chris owns two stocks, IBM and General Electric (GE). The probability that IBM stock will increase in value next year is.5 and the probability that GE 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?

26. 5- 26 Example 6 continued P(at least one) = P(IBM but not GE) + P(GE but not IBM) + P(IBM and GE) (.5)(1-.7) + (.7)(1-.5) + (.7)(.5) =.85

27. 5- 27 Conditional Probability The probability of event A occurring given that the event B has occurred is written P(A|B). Conditional Probability A Conditional Probability is the probability of a particular event occurring, given that another event has occurred.

28. 5- 28 General Multiplication Rule 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. General Rule of Multiplication The General Rule of Multiplication is used to find the joint probability that two events will occur.

29. 5- 29 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)

30. 5- 30 Example 7 The Dean of the School of Business at Owens University collected the following information about undergraduate students in her college:

31. 5- 31 Example 7 continued P(A|F) = P(A and F)/P(F) = [110/1000]/[400/1000] =.275 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?

32. 5- 32 Tree Diagrams   Example 8: In a bag containing 7 red chips and 5 blue chips you select 2 chips one after the other without replacement. Construct a tree diagram showing this information. Tree Diagram A Tree Diagram is useful for portraying conditional and joint probabilities. It is particularly useful for analyzing business decisions involving several stages.

33. 5- 33 Example 8 continued R1 B1 R2 B2 R2 B2 7/12 5/12 6/11 5/11 7/11 4/11

34. 5- 34 Some Principles of Counting Example 10: Dr. Delong has 10 shirts and 8 ties. How many shirt and tie outfits does he have? Multiplication Formula 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. (10)(8) = 80

35. 5- 35 Some Principles of Counting Permutation A Permutation is any arrangement of r objects selected from n possible objects. Note: The order of arrangement is important in permutations.

36. 5- 36 Some Principles of Counting Combination A Combination is the number of ways to choose r objects from a group of n objects without regard to order.

37. 5- 37 Example 11 There are 12 players on the Carolina Forest 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? (Order does not matter.)

38. 5- 38 Example 11 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 (order matters).