Chapter 7 Probability

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# Chapter 7 Probability - PowerPoint PPT Presentation

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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Presentation Transcript
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 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 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 Probability
• The Accumulated Relative Frequency of
• Easy Calls Converges to 70%
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 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 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 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 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 Probability
• 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 Probability
• An Example – Movie Schedule
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 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
• 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 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
• 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 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 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 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 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 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 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
• 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)
• Only add probabilities of disjoint events.