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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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Chapter 7Probability

• 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?

• 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)

• 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.

• The Accumulated Relative Frequency of

• Easy Calls Converges to 70%

• Sample Space

• Set of all possible outcomes

• Denoted by S; S = {easy, hard}

• Subsets of samples spaces are events; denoted as A, B, etc.

• Venn Diagrams

• The probability of an event A is denoted as P(A).

• Venn diagrams are graphs for depicting the relationships among events

• 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)

• 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)

• Rule 4: Complement Rule

• The complement of event A consists of the outcomes in S but not in A

• Denoted as Ac

• 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”

• An Example – Movie Schedule

• What’s the probability that the next customer

• buys a ticket for a movie that starts at 9 PM

• or is a drama?

• 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

• 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

• 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.

• 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

• 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

• 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.

• Be clear about independence.

• Only multiply probabilities of independent events.

• Do not assume that events are disjoint.

• Avoid assigning the same probability to every outcome.

• Do not confuse independent events with disjoint events.