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### Chap04 basic probability

• 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 Statisticsfor 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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