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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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Chapter 7Probability


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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