- 1. Statistics for Managers Using Microsoft® Excel 4th Edition Chapter 4 Basic Probability Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-1
- 2. Chapter Goals After completing this chapter, you should be able to: Explain basic probability concepts and definitions Use contingency tables to view a sample space Apply common rules of probability Compute conditional probabilities Determine whether events are statistically independent Use Bayes’ Theorem Statistics for Managers Using for conditional probabilities Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-2
- 3. Important Terms Probability – the chance that an uncertain event will occur (always between 0 and 1) Event – Each possible type of occurrence or outcome Simple Event – an event that can be described by a single characteristic Sample Space – the collection of all possible events Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-3
- 4. Assessing Probability There are three approaches to assessing the probability of un uncertain event: 1. a priori classical probability probability of occurrence = X number of ways the event can occur = T total number of elementary outcomes 2. empirical classical probability probability of occurrence = number of favorable outcomes observed total number of outcomes observed 3. subjective probability an individual Using Statistics for Managers judgment or opinion about the probability of occurrence Microsoft Excel, 4e © 2004 Chap 4-4 Prentice-Hall, Inc.
- 5. Sample Space The Sample Space is the collection of all possible events e.g. All 6 faces of a die: e.g. All 52 cards of a bridge deck: Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-5
- 6. Events Simple event Complement of an event A (denoted A’) An outcome from a sample space with one characteristic e.g., A red card from a deck of cards All outcomes that are not part of event A e.g., All cards that are not diamonds Joint event Involves two or more characteristics simultaneously Statisticsfor Managers that is also red from a deck of cards e.g., An ace Using Microsoft Excel, 4e © 2004 Chap 4-6 Prentice-Hall, Inc.
- 7. Visualizing Events Contingency Tables Ace Not Ace Total Black 26 2 24 26 Total Sample Space 24 Red 2 4 48 52 Tree Diagrams Full Deck of 52 Cards ar ack C Bl d Statistics for Managers Red Ca Using rd Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 2 Ac e No t a n A c e Ace No t a n Sample Space 24 2 Ace 24 Chap 4-7
- 8. Mutually Exclusive Events Mutually exclusive events Events that cannot occur together example: A = queen of diamonds; B = queen of clubs Events A and B are mutually exclusive Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-8
- 9. Collectively Exhaustive Events Collectively exhaustive events One of the events must occur The set of events covers the entire sample space example: A = aces; B = black cards; C = diamonds; D = hearts Events A, B, C and D are collectively exhaustive (but not mutually exclusive – an ace may also be a heart) Events B, C Statistics for Managersand D are collectively exhaustive and Using also mutually exclusive Microsoft Excel, 4e © 2004 Chap 4-9 Prentice-Hall, Inc.
- 10. Probability Probability is the numerical measure of the likelihood that an event will occur The probability of any event must be between 0 and 1, inclusively 0 ≤ P(A) ≤ 1 For any event A The sum of the probabilities of all mutually exclusive and collectively exhaustive events is 1 P(A) + P(B) + P(C) = 1 Statistics for Managers Using If A, B, and C are mutually exclusive and Microsoft Excel, 4e © 2004 collectively exhaustive Prentice-Hall, Inc. 1 Certain .5 0 Impossible Chap 4-10
- 11. Computing Joint and Marginal Probabilities The probability of a joint event, A and B: number of outcomes satisfying A and B P( A and B) = total number of elementary outcomes Computing a marginal (or simple) probability: P(A) = P(A and B1 ) + P(A and B 2 ) + + P(A and Bk ) Where B1, B …, Bk Statistics for Managers2,Usingare k mutually exclusive and collectively exhaustive events Microsoft Excel, 4e © 2004 Chap 4-11 Prentice-Hall, Inc.
- 12. Joint Probability Example P(Red and Ace) = number of cards that are red and ace 2 = total number of cards 52 Type Color Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Statistics forTotal Managers Using 26 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 26 52 Chap 4-12
- 13. Marginal Probability Example P(Ace) = P( Ace and Re d) + P( Ace and Black ) = Type Color 2 2 4 + = 52 52 52 Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Statistics forTotal Managers Using 26 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 26 52 Chap 4-13
- 14. Joint Probabilities Using Contingency Table Event Event B1 B2 Total A1 P(A1 and B1) P(A1 and B2) A2 P(A2 and B1) P(A2 and B2) P(A2) Total P(B1) P(B2) P(A1) 1 Joint for Managers Using Marginal (Simple) Probabilities Statistics Probabilities Microsoft Excel, 4e © 2004 Chap 4-14 Prentice-Hall, Inc.
- 15. General Addition Rule General Addition Rule: P(A or B) = P(A) + P(B) - P(A and B) If A and B are mutually exclusive, then P(A and B) = 0, so the rule can be simplified: P(A or B) = P(A) + P(B) For mutually exclusive events A and B Statistics for Managers Using Microsoft Excel, 4e © 2004 Chap 4-15 Prentice-Hall, Inc.
- 16. General Addition Rule Example P(Red or Ace) = P(Red) +P(Ace) - P(Red and Ace) = 26/52 + 4/52 - 2/52 = 28/52 Type Color Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 Don’t count the two red aces twice! 52 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-16
- 17. Computing Conditional Probabilities A conditional probability is the probability of one event, given that another event has occurred: P(A and B) P(A | B) = P(B) The conditional probability of A given that B has occurred P(A and B) P(B | A) = P(A) The conditional probability of B given that A has occurred Where P(A and B) = joint probability of A and B P(A) = marginal Statistics for Managers Using probability of A P(B) = marginal probability of B Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-17
- 18. 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. 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) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-18
- 19. Conditional Probability Example (continued) 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. CD No CD Total AC .2 .5 .7 No AC .2 .1 .3 Total .4 .6 1.0 P(CD and AC) .2 P(CD | AC) = = = .2857 Statistics for Managers Using P(AC) .7 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-19
- 20. Conditional Probability Example (continued) 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%. CD No CD Total AC .2 .5 .7 No AC .2 .1 .3 Total .4 .6 1.0 P(CD and AC) .2 P(CD | AC) = = = .2857 P(AC) .7 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-20
- 21. Using Decision Trees .2 D .7 C Has Given AC or no AC: P(A H All Cars )= . C 7 C sA a Do e hav s not eA P(A C C’) = .3 P(AC and CD) = .2 D oe s no t .5 have CD P(AC and CD’) = .5 .7 .2 D .3 C Has D P(AC’ and CD) = .2 Statistics for Managers Usingh oes not ave CD .1 P(AC’ and CD’) = .1 Microsoft Excel, 4e © 2004 .3 Chap 4-21 Prentice-Hall, Inc.
- 22. Using Decision Trees C as A H Given CD or no CD: P(C H All Cars C as )= . D 4 D Do e hav s not eC D P(C D’) = .6 .2 .4 D oe s no t .2 have AC (continued) P(CD and AC) = .2 P(CD and AC’) = .2 .4 .5 .6 C A Has D P(CD’ and AC) = .5 Statistics for Managers Usingh oes not ave AC .1 P(CD’ and AC’) = .1 Microsoft Excel, 4e © 2004 .6 Chap 4-22 Prentice-Hall, Inc.
- 23. Statistical Independence Two events are independent if and only if: P(A | B) = P(A) Events A and B are independent when the probability of one event is not affected by the other event Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-23
- 24. Multiplication Rules Multiplication rule for two events A and B: P(A and B) = P(A | B) P(B) Note: If A and B are independent, then P(A | B) = P(A) and the multiplication rule simplifies to P(A and B) = P(A) P(B) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-24
- 25. Marginal Probability Marginal probability for event A: P(A) = P(A | B1 ) P(B1 ) + P(A | B 2 ) P(B 2 ) + + P(A | Bk ) P(Bk ) Where B1, B2, …, Bk are k mutually exclusive and collectively exhaustive events Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-25
- 26. Bayes’ Theorem P(A | Bi )P(Bi ) P(Bi | A) = P(A | B1 )P(B1 ) + P(A | B 2 )P(B 2 ) + + P(A | Bk )P(Bk ) where: Bi = ith event of k mutually exclusive and collectively exhaustive events A = new event that might impact P(Bi) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-26
- 27. 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 Statistics for Managers Usingsuccessful? Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-27
- 28. Bayes’ Theorem Example (continued) Let S = successful well U = unsuccessful well P(S) = .4 , P(U) = .6 Define the detailed test event as D Conditional probabilities: P(D|S) = .6 (prior probabilities) P(D|U) = .2 Goal is to find P(S|D) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-28
- 29. Bayes’ Theorem Example (continued) Apply Bayes’ Theorem: P(D | S)P(S) P(S | D) = P(D | S)P(S) + P(D | U)P(U) (.6)(.4) = (.6)(.4) + (.2)(.6) .24 = = .667 .24 + .12 So the Managers Using Statistics forrevised probability of success, given that this well has been scheduled Microsoft Excel, 4e © 2004 for a detailed test, is .667 Chap 4-29 Prentice-Hall, Inc.
- 30. Bayes’ Theorem Example (continued) Given the detailed test, the revised probability of a successful well has risen to .667 from the original estimate of .4 Event Prior Prob. Conditional Prob. Joint Prob. Revised Prob. S (successful) .4 .6 .4*.6 = .24 .24/.36 = .667 U (unsuccessful) .6 .2 .6*.2 = .12 .12/.36 = .333 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Sum = .36 Chap 4-30
- 31. Chapter Summary Discussed basic probability concepts Examined basic probability rules Sample spaces and events, contingency tables, simple probability, and joint probability General addition rule, addition rule for mutually exclusive events, rule for collectively exhaustive events Defined conditional probability Statistical independence, marginal probability, decision trees, and the multiplication rule Discussed Bayes’ theorem Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-31

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