Statistics for business decision making and analysis robert stine and foster chapter 07

7.1 From Data to ProbabilityProbability = Long Run Relative Frequency  Keep track of calls 1 = easy call; 0 = hard call  Graph the accumulated relative frequency of easy calls  In th

Chapter 7

7.1 From Data to Probability

In a call center, what is the probability that an agent answers an easy call?

agent; a hard call needs further assistance

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

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

Note: The pattern must converge for LLN

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

7.2 Rules for Probability

Rule 1: Since S is the set of all possible

outcomes, P(S) = 1

7.2 Rules for Probability

Rule 2: For any event A, 0 ≤ P(A) ≤ 1.

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

If A and B are disjoint events, then

P (A or B) = P(A) + P(B).

7.2 Rules for Probability

Rule 3: Addition Rule for Disjoint Events

 Extends to more than two events

 P (E 1 or E 2 or … or E k ) =

P(E 1 ) + P(E 2 ) + … + P(E k )

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 A c

7.2 Rules for Probability

Rule 4: Complement Rule: P(A) = 1 –

P(A c )

The probability of an event is one minus

the probability of its complement.

7.2 Rules for Probability

Rule 5: General 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 Probability

Rule 5: General Addition Rule

P (A or B) = P(A) + P(B) – P (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

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

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

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

Best Practices

includes all of the possibilities.

 Include all of the pieces when describing

an event.

Best Practices (Continued)

 Only add probabilities of disjoint events.

 Be clear about independence.

 Only multiply probabilities of

independent events.