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- 1. Statistics for Managers Using Microsoft® Excel 4th Edition Chapter 4 Basic ProbabilityStatistics 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 independentStatistics for Managers Using for conditional probabilities Use Bayes’ TheoremMicrosoft Excel, 4e © 2004Prentice-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 eventsStatistics for Managers UsingMicrosoft Excel, 4e © 2004Prentice-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 X number of ways the event can occur probability of occurrence = = T total number of elementary outcomes 2. empirical classical probability number of favorable outcomes observed probability of occurrence = total number of outcomes observed 3. subjective probabilityStatistics for Managers judgment or opinion about the probability of occurrence an individual UsingMicrosoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-4
- 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 UsingMicrosoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-5
- 6. Events Simple event An outcome from a sample space with one characteristic e.g., A red card from a deck of cards Complement of an event A (denoted A’) All outcomes that are not part of event A e.g., All cards that are not diamonds Joint event Involves two or more characteristics simultaneouslyStatisticsfor Managers that is also red from a deck of cards e.g., An ace UsingMicrosoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-6
- 7. Visualizing Events Contingency Tables Ace Not Ace Total Black 2 24 26 Red 2 24 26 Total 4 48 52 Tree Diagrams Sample Ac e 2 Space Sample ar d Space Black C No t a n A c e 24 Full Deck of 52 Cards AceStatistics for Managers Red Ca Using 2 rdMicrosoft Excel, 4e © 2004 No t a n Ace 24Prentice-Hall, Inc. 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 exclusiveStatistics for Managers UsingMicrosoft Excel, 4e © 2004Prentice-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)Statistics for Managersand D are collectively exhaustive and Events B, C Using also mutually exclusiveMicrosoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-9
- 10. Probability Probability is the numerical measure of the likelihood that an event will 1 Certain occur The probability of any event must be between 0 and 1, inclusively 0 ≤ P(A) ≤ 1 For any event A .5 The sum of the probabilities of all mutually exclusive and collectively exhaustive events is 1 P(A) + P(B) + P(C) = 1Statistics for Managers Using 0 Impossible If A, B, and C are mutually exclusive andMicrosoft Excel, 4e © 2004 collectively exhaustivePrentice-Hall, Inc. 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 )Statistics for Managers2,Usingare k mutually exclusive and collectively Where B1, B …, Bk exhaustive eventsMicrosoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-11
- 12. Joint Probability Example P(Red and Ace) number of cards that are red and ace 2 = = total number of cards 52 Color Type Red Black Total Ace 2 2 4 Non-Ace 24 24 48Statistics forTotal 26 Managers Using 26 52Microsoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-12
- 13. Marginal Probability Example P(Ace) 2 2 4 = P( Ace and Re d) + P( Ace and Black ) = + = 52 52 52 Color Type Red Black Total Ace 2 2 4 Non-Ace 24 24 48Statistics forTotal 26 Managers Using 26 52Microsoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-13
- 14. Joint Probabilities Using Contingency Table Event Event B1 B2 Total A1 P(A1 and B1) P(A1 and B2) P(A1) A2 P(A2 and B1) P(A2 and B2) P(A2) Total P(B1) P(B2) 1 Joint for Managers Using Marginal (Simple) ProbabilitiesStatistics ProbabilitiesMicrosoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-14
- 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 BStatistics for Managers UsingMicrosoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-15
- 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 Don’t count the two red Color aces twice! Type Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52Statistics for Managers UsingMicrosoft Excel, 4e © 2004Prentice-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) The conditional P(A | B) = probability of A given P(B) that B has occurred P(A and B) The conditional P(B | A) = probability of B given P(A) that A has occurred Where P(A and B) = joint probability of A and BStatistics for Managers Using probability of A P(A) = marginalMicrosoft Excel, 4e © 2004 P(B) = marginal probability of BPrentice-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 UsingMicrosoft Excel, 4e © 2004Prentice-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) = = = .2857Statistics for Managers Using P(AC) .7Microsoft Excel, 4e © 2004Prentice-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)Statistics for Managers Using .7Microsoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-20
- 21. Using Decision Trees .2Given AC or C D .7 P(AC and CD) = .2 Hasno AC: 7 C )= . D oe P(A have s no t .5 P(AC and CD’) = .5 C CD a sA H .7AllCars Do .2 e hav s not eA C P(A C D .3 P(AC’ and CD) = .2 C’) Has = .3 DStatistics for Managers Usingh oes not CD .1 P(AC’ and CD’) = .1 aveMicrosoft Excel, 4e © 2004Prentice-Hall, Inc. .3 Chap 4-21
- 22. Using Decision Trees (continued) .2 C .4 P(CD and AC) = .2Given CD or Has Ano CD: 4 D )= . D oe P(C have s no t .2 P(CD and AC’) = .2 C D AC H as .4AllCars Do .5 e hav s not .6 eC AC P(CD’ and AC) = .5 D P(C Has D’) = .6 DStatistics for Managers Usingh oes not AC .1 P(CD’ and AC’) = .1 aveMicrosoft Excel, 4e © 2004Prentice-Hall, Inc. .6 Chap 4-22
- 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 eventStatistics for Managers UsingMicrosoft Excel, 4e © 2004Prentice-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 UsingMicrosoft Excel, 4e © 2004Prentice-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 eventsStatistics for Managers UsingMicrosoft Excel, 4e © 2004Prentice-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 UsingMicrosoft Excel, 4e © 2004Prentice-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 probabilityStatistics for Managers Usingsuccessful? that the well will beMicrosoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-27
- 28. Bayes’ Theorem Example (continued) Let S = successful well 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 Goal is to find P(S|D)Statistics for Managers UsingMicrosoft Excel, 4e © 2004Prentice-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 + .12Statistics forrevised probability of success, given that this well So the Managers UsingMicrosoft Excel, 4e © 2004 for a detailed test, is .667 has been scheduledPrentice-Hall, Inc. Chap 4-29
- 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 Conditional Joint Revised Prob. Prob. Prob. Prob. S (successful) .4 .6 .4*.6 = .24 .24/.36 = .667 U (unsuccessful) .6 .2 .6*.2 = .12 .12/.36 = .333Statistics for Managers Using Sum = .36Microsoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-30
- 31. Chapter Summary Discussed basic probability concepts Sample spaces and events, contingency tables, simple probability, and joint probability Examined basic probability rules 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’ theoremStatistics for Managers UsingMicrosoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-31

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