Chapter 7 Probability

Copyright © 2014, 2011 Pearson Education, Inc. 1

Chapter 7Probability


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?



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%


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.



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 E2 or … 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


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



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) X P(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?



Method

Past data indicates a 95% chance that the assembly line runs a full day without breaking down.



Mechanics

Assuming days are independent, use the multiplication rule to find

P (OK for 5 days) = 0.955 = 0.774



Mechanics

Use the complement rule to find

P (breakdown during 5 days) = 1 - P(OK for 5 days)= 1- 0.774 = 0.226



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 InequalityApplied to 4M Example 7.1

P (breakdown during 5 days) = P(A1 or A2 or A3 or A4 or A5)

≤ 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.

Be clear about independence.

Only multiply probabilities of independent events.


Pitfalls

Do not assume that events are disjoint.

Avoid assigning the same probability to every outcome.

Do not confuse independent events with disjoint events.

Chapter 7 Probability

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