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Chapter 7 Probability

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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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7 1 from data to 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?
7 1 from data to probability1
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)
7 1 from data to probability2
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.
7 1 from data to probability3
7.1 From Data to Probability
  • The Accumulated Relative Frequency of
  • Easy Calls Converges to 70%
7 2 rules for probability
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 2 rules for probability1
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
7 2 rules for probability4
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)
7 2 rules for probability6
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)
7 2 rules for probability7
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
7 2 rules for probability9
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”
7 2 rules for probability11
7.2 Rules for Probability
  • An Example – Movie Schedule
7 2 rules for probability12
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?
7 2 rules for probability13
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
7 3 independent events
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
7 3 independent events1
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)
4m example 7 1 managing a process
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?
4m example 7 1 managing a process1
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.
4m example 7 1 managing a process2
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
4m example 7 1 managing a process3
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
4m example 7 1 managing a process4
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.
7 3 independent events2
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
7 3 independent events4
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
best practices
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.
best practices continued
Best Practices (Continued)
  • Only add probabilities of disjoint events.
  • Be clear about independence.
  • Only multiply probabilities of independent events.
pitfalls
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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