Chapter 5

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Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

A Survey ofA Survey of

ConceptsConcepts

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When you have completed this chapter, you will be able to:

Explain the terms random experiment, outcome, sample space, permutations,

and combinations.

Define probability.

Describe the classical, empirical, and subjective

approaches to probability.

Explain and calculate conditional probability

and joint probability.

1

2

3

4

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Use a tree diagram to organize and compute probabilities.

Calculate probability using the rules of addition and rules of multiplication.

Calculate a probability using Bayes’ theorem.

5

6

7

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Types of StatisticsTypes of Statistics

Methods of… collecting

organizing presenting

and

analyzing data

Methods of… collecting

organizing presenting

and

analyzing data

DescriptiveDescriptive

Science of… making inferences about a population, based on

sample information.

Science of… making inferences about a population, based on

sample information.

InferentialInferential

Emphasis now to be on this!Emphasis now to be on this!

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TerminologyProbability

…is a measure of the likelihood that an event in the future will happen!

…is a measure of the likelihood that an event in the future will happen!

It can only assume a value between 0 and 1.

A value near zero means the event is not

likely happen; near one means it is likely..

There are three definitions of probability: classical, empirical, and subjective

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TerminologyRandom Experiment

…is a process…is a process

repetitive in nature

the outcome of any trial is uncertain

well-defined set of possible outcomes

each outcome has a probability associated with it

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…is a particular result of a

random experiment.

... is the collection or set of all the possible outcomes of a

random experiment.

Terminology

…is the collection of one or more

outcomes of an experiment.

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Approaches to Assigning Probability

SubjectiveSubjective

…probability is based on whatever information is available…probability is based on whatever information is available

ObjectiveObjective

Classical ProbabilityClassical Probability

… is based on the assumption that the

outcomes of an experiment are equally likely

… is based on the assumption that the

outcomes of an experiment are equally likely

Probability

of an Event

Probability

of an Event= NUMBER of favourable outcomes

Total NUMBER of possible outcomes

Empirical ProbabilityEmpirical Probability

… applies when the number of times the event happens is divided by the number of

observations

… applies when the number of times the event happens is divided by the number of

observations

ExamplesExamples

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…. refers to the chance of occurrence assigned to an event

by a particular individual

…. refers to the chance of occurrence assigned to an event

by a particular individual

It is not computed objectively, i.e., not from prior knowledge or from actual

data…

It is not computed objectively, i.e., not from prior knowledge or from actual

data…

S ubjectiveProbability

…that the Toronto Maple Leafs will win the Stanley Cup next season!

…that you will arrive to class on time tomorrow!

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Students measure the contents of their soft drink cans… 10 cans are underfilled,

32 are filled correctly and

8 are overfilled

Students measure the contents of their soft drink cans… 10 cans are underfilled,

32 are filled correctly and

8 are overfilled When the contents of the next can is measured,

what is the probability that it is… (a) filled correctly?

When the contents of the next can is measured, what is the probability that it is… (a) filled correctly?

P(C) = 32 / 50 = 64%…(b) not filled correctly?…(b) not filled correctly?

P(~C) = 1 – P(C) = 1 - .64 = 36%

This is called the Complement of CThis is called the Complement of C

E mpiricalProbability

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Random Experiment

The experiment is rolling the die...once!The experiment is rolling the die...once!

The possible outcomes are the numbers…

1 2 3 4 5 6

An event is the occurrence of an even number

i.e. we collect the outcomes 2, 4, and 6.

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Consider the random experiment of flipping a coin twice.

Tree Diagrams

This is a useful device to show all the possible outcomes of the experiment

and their corresponding probabilities

This is a useful device to show all the possible outcomes of the experiment

and their corresponding probabilities

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1.00

Tree Diagrams

Origin First Flip

H

T

H

TH

T

HH

HT

TT

TH Simple Events

Simple Events

P(HH)= 0.25

P(HT)= 0.25

P(TH)= 0.25

P(TT)= 0.25

SecondFlip

New

Expressed as:

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Tree Diagrams

Menu Appetizer:

Soup or Juice

Entrée:Beef

Turkey Fish

Dessert:Pie

Ice Cream

Origin Appetizer

Entrée Dessert

Soup

Juice

Beef

Turkey

Fish

Beef

Turkey

Fish

Pie

Ice CreamPie

Ice CreamPie

Ice Cream

Pie

Ice Cream

Pie

Ice Cream

PieIce

Cream

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Tree Diagrams

How many complete dinners are there?How many complete dinners are there?

5 - 12Tree Diagrams

Menu

Appetizer:Soup or J uiceEntrée:Beef

Turkey Fish

Dessert:Pie

I ce Cream

Origin Appetizer Entrée Dessert

Soup

J uice

Beef

Turkey

Fish

Beef

Turkey

Fish

Pie

I ce Cream

PieI ce Cream

Pie

I ce Cream

Pie

I ce Cream

PieI ce Cream 1212

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5 - 12Tree Diagrams

Menu


Turkey Fish

Dessert:Pie

I ce Cream


Soup

J uice

Beef

Turkey

Fish

Beef

Turkey

Fish

Pie

I ce Cream

PieI ce Cream

Pie

I ce Cream

Pie

I ce Cream

PieI ce Cream

Tree Diagrams

How many dinners include beef?How many dinners include beef?

44

1.

2.

3.

4.

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Tree Diagrams

What is the probability that a complete dinner will include…

What is the probability that a complete dinner will include…

Juice?

Turkey?

Both beef and soup?

6/126/12

4/124/12

2/122/12

5 - 12Tree Diagrams

Menu


Turkey Fish

Dessert:Pie

I ce Cream


Soup

J uice

Beef

Turkey

Fish

Beef

Turkey

Fish

Pie

I ce Cream

PieI ce Cream

Pie

I ce Cream

Pie

I ce Cream

PieI ce Cream

See next slide…

See next slide…

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If one thing can be done in M ways, and if after this is done, something else can be done in N ways, then both things can be

done in a total of M*N different ways in that stated order!

If one thing can be done in M ways, and if after this is done, something else can be done in N ways, then both things can be

done in a total of M*N different ways in that stated order!

Refer back to tree diagram example:

# different meals = 2 * 3 * 2 = 12

# meals with beef = 2 * 1 * 2 = 4

# meals with juice = 1 * 3 * 2 = 6

M * N Rule M * N Rule The

Appetizer

Entrée DessertLegend:Legend:

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3 * 2 * 5 = 30 3 * 2 * 5 = 30

When getting dressed, you have a choice between

wearing one of:3 pairs of shoes2 pairs of pants

5 shirtsFind the number of different “outfits” possible

When getting dressed, you have a choice between

wearing one of:3 pairs of shoes2 pairs of pants

5 shirtsFind the number of different “outfits” possible

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What is the probability of drawing a red Ace

from a deck of well-shuffled cards?

What is the probability of drawing a red Ace

from a deck of well-shuffled cards?

P( Red Ace) = 2/52

robability

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2.2.

1.1. Determine….the Outcomes that Meet Our Condition

List….all Possible Outcomes

Key steps

4 Suits

HeartsDiamondsClubs Spades

Deck

13 cards in each

= 52 Cards

robability

Using Analysisrobability

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P = probability …of getting four(4) aces

= 52 Cards(the Population)Deck

13 cards

4 Suitsx

4 Suits

HeartsDiamondsClubs Spades

13 cards in each

robability

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‘Honours’ cards‘Honours’ cards

Each Suit has a…….Each Suit has a…….

robability

Scenarios

= 52 Cards

4 Suits (13 cards in each)

Hearts Diamonds ClubsSpades

Deck

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52

Scenarios

1. Draw an Ace Condition Outcomes All Possible Outcomes

4

2. Draw a Black Ace Condition Outcomes All Possible Outcomes

2 52

3. Draw a Red Card Condition Outcomes All Possible Outcomes

26 52

= 52 Cards



Deck

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4. Drawing…a Red Card or a Queen

Condition Outcomes All Possible Outcomes

2652

+ 2 52

2852

=

Scenarios

= 52 Cards



Deck

-or- P(Red) + P(Queen) - P (Red Queen)

= 26 + 4 - 2

52

2852

=

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Scenarios

= 52 Cards



Deck

What is the probability of drawing a Jack or a King from a deck of well-

shuffled cards?

What is the probability of drawing a Jack or a King from a deck of well-

shuffled cards?

= 4/52

= 4/52

= 8/52= 8/52P( Jack or King) = 4/52 + 4/52

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Scenarios

What is the probability of drawing one card that is both a Jack and a King from a deck of

well-shuffled cards?

What is the probability of drawing one card that is both a Jack and a King from a deck of

well-shuffled cards?

These are MUTUALLY EXCLUSIVE events, i.e. they can’t both happen at the same time!

= 0= 0P( Jack and King)

= 52 Cards



Deck

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Scenarios

=2/52=2/52P( Black and King)

What is the probability of drawing one card that is both BLACK and a King from a deck

of well-shuffled cards?

What is the probability of drawing one card that is both BLACK and a King from a deck


= 52 Cards



Deck

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Scenarios

= 52 Cards



Deck

What is the probability of drawing a card that is either BLACK or a King from a deck


What is the probability of drawing a card that is either BLACK or a King from a deck


Formula Formula P(A or B) =

= 28/52= 28/52P( Black or King) = 26/52 + 4/52 - 2/52

This is called the Addition Rule

P (A) + P(B) –P(Both)

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Scenarios

= 52 Cards



Deck

What is the probability of drawing a King given that you have drawn a BLACK card?What is the probability of drawing a King given that you have drawn a BLACK card?

= 2/26= 2/26P(King|Black )

This is called a CONDITIONAL probability

Our sample space is now just the BLACK cards

Alternate solutionAlternate solution

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= 52 Cards



Deck

Scenarios

What is the probability of drawing a King given that you have drawn a BLACK card?What is the probability of drawing a King given that you have drawn a BLACK card?

Formula Formula P(A|B) =P(Given)P (Both)

= 2/26= 2/26

= (2/52) / (26/52)

= (2/52) * (52/26)

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Scenarios

= 52 Cards



Deck

Scenarios

= 52 Cards



Deck

What is the probability of drawing a King of Clubs given that you have drawn a BLACK

card?

What is the probability of drawing a King of Clubs given that you have drawn a BLACK

card?

P(King of Clubs|Black )

= 1/26= 1/26

= (1/52) / (26/52)

= (1/52) * (52/26)

P(A|B) =P(Given)P (Both)

Formula Formula

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Scenarios

= 52 Cards



Deck

What is the probability of drawing a King of Clubs given that you have drawn a CLUB?

What is the probability of drawing a King of Clubs given that you have drawn a CLUB?

P(King of Clubs given Club)

= 1/13= 1/13

= (1/52) * (52/13)= P(1/52) / (13/52)

= P(King of Clubs|Club)

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Reading Probabilities

from a Table

Reading Probabilities

from a Table

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What is the Probability of selecting a female student?What is the Probability of selecting a female student?

400/750 = 53.33%

400/750 = 53.33%

A survey of undergraduate students in the School of Business Management at Eton College revealed the

following regarding the gender and majors of the students: Gender Accounting International HR TOTAL

Male 150 150

50 350

Female 175 160

65 400

325 310 115 750

More

Reading Probabilities from a Table

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What is the Probability of selecting a Human Resources or International major?

What is the Probability of selecting a Human Resources or International major?

Gender Accounting International HR TOTAL

Male 150 150

50 350

Female 175 160

65 400

325 310 115 750

More

= 115/750 + 310/750 = 425/750= 56.67%

= 115/750 + 310/750 = 425/750= 56.67%

P(HR or I) = P(HR) + P(I)

310 115


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What is the Probability of selecting a Female

or International major?


or International major?


Male 150 150

50 350

Female 175 160

65 400

325 310 115 750

= 400/750

= 400/750

P(F or I) = P(F) + P(I) – P(F and I)

More


+ 310/750 – 160/750

= 550/750 = 73.33%

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Accounting student?


Accounting student?


Male 150 150

50 350

Female 175 160

65 400

325 310 115 750= 175/750 = 23.33%= 175/750 = 23.33%P(F and A)

More


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What is the Probability of selecting a Female, given that the person selected

is an International major?




Male 150 150

50 350

Female 175 160

65 400

325 310 115 750160/310 = 51.6%160/310 = 51.6%P(F|I) =

Alternative Solution


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= (160/750) / (310/750)= (160/750) / (310/750)

P(F|I) = P(F and I) / P(I)

P(A|B) =Formula Formula P(Both)P(Given)


= 51.6%= 51.6%

= 160/310

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What is the Probability of selecting an International major, given that the person

selected is a Female?

What is the Probability of selecting an International major, given that the person

selected is a Female?


Male 150 150

50 350

Female 175 160

65 400

325 310 115 750160/400 = 40%160/400 = 40%P(I|F) =

More


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Notice the significant difference:Notice the significant difference:


…between F given I …I given F!and

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Each flip is independent of the other!

Each flip is independent of the other!

Flip once

Flip twice

Terminology

Events are independent if the occurrence of

one event does not affect the probability of the other

Events are independent if the occurrence of

one event does not affect the probability of the other

Consider the random experiment of flipping a coin twice.

Independent Events

Find the probability of flipping 2 Heads in

a row

Find the probability of flipping 2 Heads in

a row

P(2H) = .5*.5 = .25 or 25%

P(2H) = .5*.5 = .25 or 25%

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TerminologyIndependent Events

Each draw is independent of the otherEach draw is independent of the other

Draw three cards with replacement i.e., draw one card, look at

it, put it back,

and repeat twice more.

Draw three cards with replacement i.e., draw one card, look at

it, put it back,

and repeat twice more.

Find the probability of drawing 3 Queens in a row:

P(3Q) = 4/52 * 4/52 *4/52P(3Q) = 4/52 * 4/52 *4/52 = 0.00046 = most unlikely!= 0.00046 = most unlikely!

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Independent Events

Consider 2 events:

Drawing a RED card from a deck of cards

Drawing a HEART from a deck of cards

Consider 2 events:

Drawing a RED card from a deck of cards

Drawing a HEART from a deck of cards

Are these two events considered to be independent?

If two events, A and B are independent, then P(A|B) = P(A)

If two events, A and B are independent, then P(A|B) = P(A)

P(Red) =

P(Red|Heart) =

26/52 = 1/2

13/13 = 1

Therefore these are NOT independent events!

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ayes’heorem

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…is a method for revising a probability given additional information!

…is a method for revising a probability given additional information!

Formula Formula

Example

ayes’heorem

P(A1|B) =P(A1 ) P(B|A1 )

P(A1 ) P(B|A1)+ P(A2 )P(B|A2 )

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ayes’heorem

Duff Cola Company recently received several complaints that their bottles are under-filled.

A complaint was received today but the production manager is unable to identify which of the two

Springfield plants (A or B) filled this bottle.

What is the probability that the under-filled bottle came from plant A?


% of Total Production % of Underfilled Bottles

A 55 3

B 45 4

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ayes’heorem

% of Total Production % of Underfilled Bottles

A 55 3

B 45 4



1 List the

Probabilities given

List the

Probabilities given

2 Input values into formula and compute

Input values into formula and compute

P(plant A) = .55

P(plant B) = .45

P(Underfilled -A) = .03

P(Underfilled -B) = .04

P(plant A) = .55

P(plant B) = .45

P(Underfilled -A) = .03

P(Underfilled -B) = .04

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ayes’heorem



1 List the

Probabilities given

List the

Probabilities given

2 Input values into formula and compute

Input values into formula and compute

P(plant A) = .55

P(plant B) = .45

P(Underfilled/A) = .03

P(Underfilled/B) = .04

P(plant A) = .55

P(plant B) = .45

P(Underfilled/A) = .03

P(Underfilled/B) = .04

P(A1 |B) =P(A1 ) P(B|A1 )

P(A1 )P(B|A1 )+ P(A2 ) P(B|A2 )

= .55(.03)

.55(.03) + .45(.04) = .4783

= .4783

The likelihood that the underfilled bottle came from Plant A has been reduced from

55% to 47.83%

The likelihood that the underfilled bottle came from Plant A has been reduced from

55% to 47.83%

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Counting

Rules

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actorials! … this is just a shorthand notation

that is sometimes used to save time!

Examples:5! … Means 5*4*3*2*1 = 1204! … Means 4*3*2*1 = 24

Examples:5! … Means 5*4*3*2*1 = 1204! … Means 4*3*2*1 = 24

By definition, 1! =1 and 0! =1By definition, 1! =1 and 0! =1

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…is a counting technique

that is used when order is important!


that is used when order is important!


that is used when order is NOT important!…is a counting technique

that is used when order is NOT important!

ermutation

ombination

n Pr =n!

(n – r)!

n Cr =n!

r!(n – r)!

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…How many ways can you arrange n things, taking r at a time, when order is important?

…How many ways can you arrange n things, taking r at a time, when order is important?

You are assigned the task of choosing 2 of your 6 classmates to serve on a task force. One will act as the

Chair of the task force, and the other will be the Secretary. In how many

ways can you make this assignment?

You are assigned the task of choosing 2 of your 6 classmates to serve on a task force. One will act as the

Chair of the task force, and the other will be the Secretary. In how many

ways can you make this assignment?

ermutation

n Pr =n!

(n – r)!

Example:Example:

6P2 = 6! / (6-2)! = 6! / 4! = 6*5 = 30

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You are assigned the task of choosing 2 of your 6 classmates to serve on a task force. Responsibilities are evenly shared.

In how many ways can you make this assignment?

You are assigned the task of choosing 2 of your 6 classmates to serve on a task force. Responsibilities are evenly shared.

In how many ways can you make this assignment?

Example:Example:

6C2 = 6! / (2!(6-2)!) = 6! /2!4! = (6*5)/2 = 15

…is a counting technique that is used when order is

NOT important!

…is a counting technique that is used when order is

NOT important!

ombination

n Cr =n!

r(n – r)!

Using…

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Using… Texas Instruments BAII PLUS

i

15 30 ombination ermutation

nCr

6

2

1515

6

2

3030

nPr

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Test your learning…Test your learning…

www.mcgrawhill.ca/college/lindClick on…Click on…

Online Learning Centrefor quizzes

extra contentdata setssearchable glossaryaccess to Statistics Canada’s E-Stat data…and much more!

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This completes Chapter 5This completes Chapter 5

Chapter 5

Documents

mcgrawhill companies

probability of drawing

p black

red card copyright

p jack

black ace

p red ace

deck of wellshuffled