Chap04 basic probability

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Chap04 basic probability

  1. 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. 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. 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. 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. 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. 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 simultaneouslyStatisticsfor Managers that is also red from a deck of cards e.g., An ace UsingMicrosoft Excel, 4e © 2004Prentice-Hall, Inc. Chap 4-6
  7. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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