1. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-1 Business Statistics: A Decision-Making Approach 7 th Edition Chapter 4 Using Probability and Probability Distributions

2. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-2 Chapter Goals After completing this chapter, you should be able to: Explain three approaches to assessing probabilities Apply common rules of probability, including the Addition Rule and the Multiplication Rule Use Bayes’ Theorem for conditional probabilities

3. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-3 Important Terms Probability – the chance that an uncertain event will occur (always between 0 and 1) Experiment – a process of obtaining outcomes for uncertain events Experimental Outcome – the most basic outcome possible from a simple experiment Sample Space – the collection of all possible experimental outcomes

4. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-4 Sample Space The Sample Space is the collection of all possible outcomes e.g., All 6 faces of a die: e.g., All 52 cards of a bridge deck:

5. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-5 Events Experimental outcome – An outcome from a sample space with one characteristic Example: A red card from a deck of cards Event – May involve two or more outcomes simultaneously Example: An ace that is also red from a deck of cards

6. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-6 Visualizing Events Contingency Tables Tree Diagrams Red 2 24 26 Black 2 24 26 Total 4 48 52 Ace Not Ace Total Full Deck of 52 Cards Red Card Black Card Not an Ace Ace Not an Ace Sample Space 2 24 2 24

7. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-7 Experimental Outcomes A automobile consultant records fuel type and vehicle type for a sample of vehicles 2 Fuel types: Gasoline, Diesel 3 Vehicle types: Truck, Car, SUV 6 possible experimental outcomes: e 1 Gasoline, Truck e 2 Gasoline, Car e 3 Gasoline, SUV e 4 Diesel, Truck e 5 Diesel, Car e 6 Diesel, SUV Gasoline Diesel Car Truck Car SUV e1e2e3e4e5e6e1e2e3e4e5e6

8. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-8 Probability Concepts Mutually Exclusive Events If E 1 occurs, then E 2 cannot occur E 1 and E 2 have no common elements Black Cards Red Cards A card cannot be Black and Red at the same time. E1E1 E2E2

9. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-9 Independent and Dependent Events Independent: Occurrence of one does not influence the probability of occurrence of the other Dependent: Occurrence of one affects the probability of the other Probability Concepts

10. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-10 Independent Events E 1 = heads on one flip of fair coin E 2 = heads on second flip of same coin Result of second flip does not depend on the result of the first flip. Dependent Events E 1 = rain forecasted on the news E 2 = take umbrella to work Probability of the second event is affected by the occurrence of the first event Independent vs. Dependent Events

11. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-11 Assigning Probability Classical Probability Assessment Relative Frequency of Occurrence Subjective Probability Assessment P(E i ) = Number of ways E i can occur Total number of experimental outcomes Relative Freq. of E i = Number of times E i occurs N An opinion or judgment by a decision maker about the likelihood of an event

12. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-12 Rules of Probability Rules for Possible Values and Sum Individual ValuesSum of All Values 0 ≤ P(E i ) ≤ 1 For any event E i where: k = Number of individual outcomes in the sample space e i = i th individual outcome Rule 1Rule 2

13. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-13 Addition Rule for Elementary Events The probability of an event E i is equal to the sum of the probabilities of the individual outcomes forming E i. That is, if: E i = {e 1, e 2, e 3 } then: P(E i ) = P(e 1 ) + P(e 2 ) + P(e 3 ) Rule 3

14. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-14 Complement Rule The complement of an event E is the collection of all possible elementary events not contained in event E. The complement of event E is represented by E. Complement Rule: E E Or,

15. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-15 Addition Rule for Two Events P(E 1 or E 2 ) = P(E 1 ) + P(E 2 ) - P(E 1 and E 2 ) E1E1 E2E2 Don’t count common elements twice! ■ Addition Rule: E1E1 E2E2 += Rule 4

16. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-16 Addition Rule Example P(Red or Ace) = P(Red) +P(Ace) - P(Red and Ace) = 26/52 + 4/52 - 2/52 = 28/52 Don’t count the two red aces twice! Black Color Type Red Total Ace 224 Non-Ace 24 48 Total 26 52

17. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-17 Addition Rule for Mutually Exclusive Events If E 1 and E 2 are mutually exclusive, then P(E 1 and E 2 ) = 0 So P(E 1 or E 2 ) = P(E 1 ) + P(E 2 ) - P(E 1 and E 2 ) P(E 1 or E 2 ) = P(E 1 ) + P(E 2 ) = 0 E1E1 E2E2 if mutually exclusive Rule 5

18. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-18 Conditional Probability Conditional probability for any two events E 1, E 2 : Rule 6

19. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-19 What is the probability that a car has a CD player, given that it has AC ? i.e., we want to find P(CD | AC) Conditional Probability Example Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.

20. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-20 Conditional Probability Example No CDCDTotal AC.2.5.7 No AC.2.1.3 Total.4.61.0 Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both. (continued)

21. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-21 Conditional Probability Example No CDCDTotal AC.2.5.7 No AC.2.1.3 Total.4.6 1.0 Given AC, we only consider the top row (70% of the cars). Of these, 20% have a CD player. 20% of 70% is about 28.57%. (continued)

22. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-22 For Independent Events: Conditional probability for independent events E 1, E 2 : Rule 7

23. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-23 Multiplication Rules Multiplication rule for two events E 1 and E 2 : Note: If E 1 and E 2 are independent, then and the multiplication rule simplifies to Rule 8 Rule 9

24. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-24 Tree Diagram Example Diesel P(E 2 ) = 0.2 Gasoline P(E 1 ) = 0.8 Truck: P(E 3 |E 1 ) = 0.2 Car: P(E 4 |E 1 ) = 0.5 SUV: P(E 5 |E 1 ) = 0.3 P(E 1 and E 3 ) = 0.8 x 0.2 = 0.16 P(E 1 and E 4 ) = 0.8 x 0.5 = 0.40 P(E 1 and E 5 ) = 0.8 x 0.3 = 0.24 P(E 2 and E 3 ) = 0.2 x 0.6 = 0.12 P(E 2 and E 4 ) = 0.2 x 0.1 = 0.02 P(E 3 and E 4 ) = 0.2 x 0.3 = 0.06 Truck: P(E 3 |E 2 ) = 0.6 Car: P(E 4 |E 2 ) = 0.1 SUV: P(E 5 |E 2 ) = 0.3

25. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-25 Bayes’ Theorem where: E i = i th event of interest of the k possible events B = new event that might impact P(E i ) Events E 1 to E k are mutually exclusive and collectively exhaustive

26. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-26 Bayes’ Theorem Example A drilling company has estimated a 40% chance of striking oil for their new well. A detailed test has been scheduled for more information. Historically, 60% of successful wells have had detailed tests, and 20% of unsuccessful wells have had detailed tests. Given that this well has been scheduled for a detailed test, what is the probability that the well will be successful?

27. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-27 Let S = successful well and U = unsuccessful well P(S) =.4, P(U) =.6 (prior probabilities) Define the detailed test event as D Conditional probabilities: P(D|S) =.6 P(D|U) =.2 Revised probabilities Bayes’ Theorem Example Event Prior Prob. Conditional Prob. Joint Prob. Revised Prob. S (successful).4.6.4*.6 =.24.24/.36 =.67 U (unsuccessful).6.2.6*.2 =.12.12/.36 =.33 Sum =.36 (continued)

28. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-28 Given the detailed test, the revised probability of a successful well has risen to.67 from the original estimate of.4 Bayes’ Theorem Example Event Prior Prob. Conditional Prob. Joint Prob. Revised Prob. S (successful).4.6.4*.6 =.24.24/.36 =.67 U (unsuccessful).6.2.6*.2 =.12.12/.36 =.33 Sum =.36 (continued)

29. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-29 Chapter Summary Described approaches to assessing probabilities Developed common rules of probability Addition Rules Multiplication Rules Defined conditional probability Used Bayes’ Theorem for conditional probabilities