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Chapter 4

Chapter 4. Probability. Probability Defined. A probability is a number between 0 and 1 that measures the chance or likelihood that some event or set of events will occur. Assigning Basic Probabilities. Classical Approach Relative Frequency Approach Subjective Approach.

norman-combs
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Chapter 4

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  1. Chapter 4 Probability

  2. Probability Defined A probability is a number between 0 and 1 that measures the chance or likelihood that some event or set of events will occur.

  3. Assigning Basic Probabilities • Classical Approach • Relative Frequency Approach • Subjective Approach

  4. Classical Approach P(A) = where P(A) = probability of event A F = number of outcomes “favorable” to event A T = total number of outcomes possible in the experiment

  5. Relative Frequency Approach P(A) = where N = total number of observations or trials n = number of times that event A occurs

  6. The Language of Probability • Simple Probability • Conditional Probability • Independent Events • Joint Probability • Mutually Exclusive Events • Either/Or Probability

  7. Statistical Independence Two events are said to be statistically independent if the occurrence of one event has no influence on the likelihood of occurrence of the other.

  8. Statistical Independence (4.1) P(A l B) = P(A) and P (B l A) = P(B) “given” “given”

  9. General Multiplication Rule(4.2) P(AB) = P(A)∙ P(BlA) “and”

  10. Multiplication Rule for (4.3) Independent Events P(AB) = P(A) ∙P(B)

  11. Mutually Exclusive Events Two events, A and B, are said to be mutually exclusive if the occurrence of one event means that the other event cannot or will not occur.

  12. Mutually Exclusive Events (4.4) P(AB) = 0

  13. General Addition Rule (4.5) P(AB) = P(A) + P(B) - P(AB) “or”

  14. Addition Rule for (4.6) Mutually Exclusive Events P(AB) = P(A) + P(B)

  15. “Conditional Equals JOINT Over SIMPLE” Rule P(BIA) = (4.7a) P(AIB) = (4.7b)

  16. Complementary Events Rule (4.8) P(A′ ) = 1 – P(A)

  17. Figure 4.1 Venn Diagram for the Internet Shoppers Example Sample Space (1.0) A(.8) (Airline Ticket Purchase) B(.6) (Book Purchase) A∩ B (.5) The sample space contains 100% of the possible outcomes in the experiment. 80% of these outcomes are in Circle A; 60% are in Circle B; 50% are in both circles.

  18. Figure 4.2 Complementary Events Sample Space (1.0) The events A and A’ are said to be complementary since one or the other (but never both) must occur. For such events, P(A’ ) = 1 - P(A). A A’ (.2) (everything in the sample space outside A) .8

  19. Figure 4.3 Mutually Exclusive Events Mutually exclusive events appear as non-overlapping circles in a Venn diagram. Sample Space (1.0) A B

  20. Figure 4.4 Probability Tree for the Project Example STAGE 1 STAGE 2 PROJECT B PROJECT A PERFORMANCE PERFORMANCE B A B is under budget A is under budget B' B is not under budget B B is under budget A' B' A is not under budget B is not under budget

  21. STAGE 1 STAGE 2 PROJECT B PROJECT A PERFORMANCE PERFORMANCE B(.6) A (.25) B is under budget A is under budget B‘(.4) B is not under budget B(.2) B is under budget A‘(.75) B‘(.8) A is not under budget B is not under budget Figure 4.5 Showing Probabilities on the Tree

  22. STAGE 1 STAGE 2 PROJECT B PROJECT A PERFORMANCE PERFORMANCE B(.6) (1) A (.25) B is under budget A is under budget B‘(.4) (2) B is not under budget B(.2) (3) A‘(.75) B is under budget B‘(.8) A is not under budget (4) B is not under budget Figure 4.6 Identifying the Relevant End Nodes On The Tree  

  23. STAGE 1 STAGE 2 PROJECT B PROJECT A PERFORMANCE PERFORMANCE B(.6) (1) A (.25) B is under budget A is under budget B‘(.4) (2) B is not under budget B(.2) (3) B is under budget A‘(.75) B‘(.8) A is not under budget (4) B is not under budget Figure 4.7 Calculating End Node Probabilities .10 .15

  24. SOURCE CONDITION B (.04) A (.7) Unit is defective 1 B' (.96) Adams is the supplier Unit is OK B (.07) A (.3) Unit is defective 2 B' (.93) Alder is the supplier Unit is OK Figure 4.8 Probability Tree for the Spare Parts Example

  25. Figure 4.9 Using the Tree to Calculate End-Node Probabilities SOURCE CONDITION B (.04)  .028 (.7) A Unit is defective 1 Adams is the supplier B' (.96) Unit is OK B (.07)  .021 (.3) A Unit is defective 2 B' (.93) Alder is the supplier Unit is OK

  26. Bayes’ Theorem (4.9) (Two Events) P(A1lB) =

  27. General Form of Bayes’ Theorem P(AilB) =

  28. Cross-tabs Table

  29. Joint Probability Table

  30. Counting Total Outcomes (4.10) in a Multi-Stage Experiment Total Outcomes = m1 x m2 x m3 x…x mk where mi = number of outcomes possible in each stage k = number of stages

  31. Combinations (4.11) nCx= where nCx= number of combinations (subgroups) of n objects selected x at a time n = size of the larger group x = size of the smaller subgroups

  32. 2 A B E C D 3 4 Figure 4.10 Your Car and Your Friends Five friends are waiting for a ride in your car, but only four seats are available. How many different arrangements of friends in the car are possible? Friends 1 Car

  33. Permutations (4.12) nPx=

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