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Chapter 7 Probability

Chapter 7 Probability. 7.1 From Data to Probability. In a call center, what is the probability that an agent answers an easy call? An easy call can be handled by a first-tier agent; a hard call needs further assistance Two possible outcomes: easy and hard calls Are they equally likely?.

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Chapter 7 Probability

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  1. Chapter 7Probability

  2. 7.1 From Data to Probability • In a call center, what is the probability that an • agent answers an easy call? • An easy call can be handled by a first-tier agent; a hard call needs further assistance • Two possible outcomes: easy and hard calls • Are they equally likely?

  3. 7.1 From Data to Probability • Probability = Long Run Relative Frequency • Keep track of calls (1 = easy call; 0 = hard call) • Graph the accumulated relative frequency of easy calls • In the long run, the accumulated relative frequency converges to a constant (probability)

  4. 7.1 From Data to Probability • The Law of Large Numbers (LLN) • The relative frequency of an outcome • converges to a number, the probability of the • outcome, as the number of observed • outcomes increases. • Notes: The pattern must converge for LLN to apply. LLN only applies in the long run.

  5. 7.1 From Data to Probability • The Accumulated Relative Frequency of • Easy Calls Converges to 70%

  6. 7.2 Rules for Probability • Sample Space • Set of all possible outcomes • Denoted by S; S = {easy, hard} • Subsets of samples spaces are events; denoted as A, B, etc.

  7. 7.2 Rules for Probability • Venn Diagrams • The probability of an event A is denoted as P(A). • Venn diagrams are graphs for depicting the relationships among events

  8. 7.2 Rules for Probability

  9. 7.2 Rules for Probability

  10. 7.2 Rules for Probability • Rule 3: Addition Rule for Disjoint Events • Disjoint events are mutually exclusive; • i.e., they have no outcomes in common. • The union of two events is the collection of outcomes in A, in B, or in both (A or B)

  11. 7.2 Rules for Probability

  12. 7.2 Rules for Probability • Rule 3: Addition Rule for Disjoint Events • Extends to more than two events • P (E1 or E2or … or Ek) = • P(E1) + P(E2) + … + P(Ek)

  13. 7.2 Rules for Probability • Rule 4: Complement Rule • The complement of event A consists of the outcomes in S but not in A • Denoted as Ac

  14. 7.2 Rules for Probability

  15. 7.2 Rules for Probability • Rule 5: Addition Rule • The intersection of A and B contains the outcomes in both A and B • Denoted as A ∩ B read “A and B”

  16. 7.2 Rules for Probability

  17. 7.2 Rules for Probability • An Example – Movie Schedule

  18. 7.2 Rules for Probability • What’s the probability that the next customer • buys a ticket for a movie that starts at 9 PM • or is a drama?

  19. 7.2 Rules for Probability • What’s the probability that the next customer • buys a ticket for a movie that starts at 9 PM • or is a drama? • Use the General Addition Rule: • P(A or B) = P(9 PM or Drama) • = 3/6 + 3/6 – 2/6 • = 2/3

  20. 7.3 Independent Events • Definitions • Two events are independent if the occurrence of one does not affect the chances for the occurrence of the other • Events that are not independent are called dependent

  21. 7.3 Independent Events • Multiplication Rule • Two events A and B are independent if the • probability that both A and B occur is the • product of the probabilities of the two events. • P (A and B) = P(A) XP(B)

  22. 4M Example 7.1: MANAGING A PROCESS • Motivation • What is the probability that a breakdown on an assembly line will occur in the next five days, interfering with the completion of an order?

  23. 4M Example 7.1: MANAGING A PROCESS • Method • Past data indicates a 95% chance that the • assembly line runs a full day without breaking • down.

  24. 4M Example 7.1: MANAGING A PROCESS • Mechanics • Assuming days are independent, use the • multiplication rule to find • P (OK for 5 days) = 0.955 = 0.774

  25. 4M Example 7.1: MANAGING A PROCESS • Mechanics • Use the complement rule to find • P (breakdown during 5 days) • =1 - P(OK for 5 days) • = 1- 0.774 = 0.226

  26. 4M Example 7.1: MANAGING A PROCESS • Message • The probability that a breakdown interrupts • production in the next five days is 0.226. It is wise • to warn the customer that delivery may be delayed.

  27. 7.3 Independent Events • Boole’s Inequality • Also known as Bonferroni’s inequality • The probability of a union is less than or equal to the sum of the probabilities of the events

  28. 7.3 Independent Events

  29. 7.3 Independent Events • Boole’s Inequality • Applied to 4M Example 7.1 • P (breakdown during 5 days) • =P(A1orA2orA3orA4orA5) • ≤ 0.05 + 0.05 + 0.05 + 0.05 + 0.05 • ≤ 0.25 • Exact answer if the events are independent is 0.226

  30. Best Practices • Make sure that your sample space includes all of the possibilities. • Include all of the pieces when describing an event. • Check that the probabilities assigned to all of the possible outcomes add up to 1.

  31. Best Practices (Continued) • Only add probabilities of disjoint events. • Be clear about independence. • Only multiply probabilities of independent events.

  32. Pitfalls • Do not assume that events are disjoint. • Avoid assigning the same probability to every outcome. • Do not confuse independent events with disjoint events.

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