Ch 5.1 to 5.3 Copyright of the definitions and examples is reserved to Pearson Education, Inc.. In...

Los Angeles Mission College

Ch 5.1 to 5.3

Copyright of the definitions and examples is reserved to Pearson Education, Inc.. In order to use this PowerPoint presentation, the required textbook for the class is the Fundamentals of Statistics, Informed Decisions Using Data, Michael Sullivan, III, fourth edition.

Presented by DW


Section 5.1 Probability RuleObjective A : Sample Spaces and EventsObjective B : Requirements for ProbabilitiesObjective C : Calculating Probabilities

Section 5.2 The Additional Rules and ComplementsObjective A: Addition Rule for Disjoint (Mutually Exclusive) EventsObjective B : General Addition RuleObjective C : Complement RuleObjective D : Contingency Table

Section 5.3 Independence and Multiplication RuleObjective A : Independent EventsObjective B : Multiplication Rule for Independent EventsObjective C : At-Least Probabilities

Presented by DW


Section 5.1 Probability Rule

Experiment –

Objective A : Sample Spaces and Events

Outcome –

Sample Space, –

Event, –

Simple event, –

Compound event –

S

E

ie

any activity that leads to well-defined results called

outcomes.the result of a single trial of probability experiment.

the set of all possible outcomes of a probability experiment.

a subset of sample space.

an event with one outcome is called a simple event.

consists of two or more outcomes.

Presented by DW


Example 1 : A die is tossed one time. (a) List the elements of the sample space . (b) List the elements of the event consisting of a number that is greater than 4.

Example 2 : A coin is tossed twice. List the elements of the sample space , and list the elements of the event consisting of at least one head.

(a) }6,5,4,3,2,1{S

(b) }6,5{E

},,,{ TTTHHTHHS

},,{ HHTHHTE

S

H

T

H

HT

T

S

Presented by DW





Presented by DW


Objective B : Requirements for Probabilities

1. Each probability must lie on between 0 and 1.

2. The sum of probabilities for all simple events in equals 1.

If an event is impossible, the probability of the event is 0.

Probabilities should be expressed as reduced fractions or rounded to three decimal places.

)1)(0( EP

S)1)(( ieP

If an event is a certainty, the probability of the event is 1.

An unusual event is an event that has a low probability of occurring. Typically, an event with a probability less than 0.05 is considered as unusual.

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Example 1 : A probability experiment is conducted. Which of these can

be considered a probability of an outcome?52(a) (b) (c)28.0 09.1

Yes)1)(0( EP

No No)1)(0( EP

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Example 2 : Why is the following not a probability model?

Color Probability

Red 0.28

Green 0.56

Yellow 0.37

Condition 1 : )1)(0( EP

Condition 2 : )1)(( ieP

121.137.056.028.0 Check :

Condition 2 was not met.

Presented by DW


1 2 3 4{ , , , }S e e e eExample 3 : Given :

Find : 1 2( ) ( ) 0.2P e P e

3( ) 0.5P e and )( 4eP

Condition 1 : )1)(0( EP

Condition 2 : )1)(( ieP

1 2 3 4( ) ( ) ( ) ( ) 1P e P e P e P e

40.2 0.2 0.5 ( ) 1P e

40.9 ( ) 1P e

4( ) 1 0.9P e

4( ) 0.1P e

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Presented by DW


The probability of event is approximately the number of times event is observed divided by the number of repetitions of the experiment.

Objective C : Calculating ProbabilitiesC1. Approximating Probabilities Using the Empirical Approach (Relative Frequency Approximation of Probability)

ExperimentofTrialsofNumber

ofFrequencyofFrequencyRelative)(

EEEP

EE

Example 1 : Suppose that you roll a die 100 times and get six 80 times. Based on these results, what is the probability that the next roll results in six?

)six(P 100

808.0or

5

4

Presented by DW


Example 2 : During a sale at men’s store, 16 white sweaters, 3 red sweaters, 9 blue sweaters and 7 yellow sweaters were purchased. If a customer is selected at random, find the probability that he bought a sweater that was not white.

)whitenot(P

19793whitenotofFrequency

3579316trialsofNumber f

)placesdecimal3(543.0or35

19

Presented by DW


Age # of Employees

Under 20 25

20 – 29 48

30 – 39 32

40 – 49 15

50 and over 10

Example 3 : The age distribution of employees for this college is shown below:

If an employee is selected at random, find the probability that he or she is in the following age groups.

(a) Between 30 and 39 years of age

(b) Under 20 or over 49 years of age

130

32

130

1025

)placesdecimal3(246.0or65

16

130

35)placesdecimal3(269.0or

26

7

130f

Presented by DW


C2. Classical Approach to Probability (Equally Likely Outcomes are required)

If an experiment has equally likely outcomes and if the number of ways that an event can occur in , then the probability of ,

, is

If is the sample space of this experiment,

where is the number of outcomes in event , and is the number of outcomes in the sample space.

S

E

)(

)()(

SN

ENEP

mn

n

mEEP

outcomespossibleofNumber

occurcanthatwaysofNumber)(

E)(EP

)(SN)(EN E

Presented by DW


(b) Compute the probability of the event “an odd number.”

Example 1 : Let the sample space be . Suppose the outcomes are equally likely.

}10,9,8,7,6,5,4,3,2,1{S

)(

)()(

SN

FNFP

2)( FN

(a) Compute the probability of the event .

10)( SN

}9,5{F

E

{1, 3, 5, 7, 9}E 5)( EN

)(

)()(

SN

ENEP

10

22.0or

5

1

10

55.0or

2

1

Presented by DW


Example 2 : Two dice are tossed. Find the probability that the sum of two dice is greater than 8.

)(

)()(

SN

ENEP

36)( SN 10)( EN

)}6,6(),5,6(),4,6(),3,6(),6,5(

),5,5(),4,5(),6,4(),5,4(),6,3{(E 36

10278.0or

18

5

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Example 3 : If one card is drawn from a deck, find the probability of getting (a) a club; (b) a 4 and a club.

)(

)club (a)cluba(

SN

NP

(a) a club

52

1325.0or

4

1

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)(

)clubaand4a()clubaand4a(

SN

NP

(b) a 4 and a club

019.052

1

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)(

)()(

SN

ENEP

},,,,,{ ABMAMBBAMBMAMABMBAS (a)

Example 4 : Three equally qualified runners, Mark, Bill, and Alan, run a 100-meter sprint, and the order of finish is recorded. (a) Give a sample space . (b) What is the probability that Mark will finish last?

(b) },{ ABMBAME

S

M

M

M

M

B

B

B

B

A

A

A

A

M

A

B

6

2333.0or

3

1

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Presented by DW


Section 5.2 The Additional Rules and Complements

Objective A: Addition Rule for Disjoint (Mutually Exclusive) Events

Event A and B are disjoint (mutually exclusive) if they have no outcomes in common.

Addition Rule for Disjoint Events

If E and F are disjoint events, then .)()()or( FPEPFEP

Presented by DW


Example 1: A standard deck of cards contain 52 cards. One card is randomly selected from the deck. Compute the probability of randomly selecting a two or three from a deck of cards.

)3()2()3or2( PPP 52

4

52

4

52

8154.0

13

2or

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Presented by DW


Objective B : General Addition Rule

The General Addition Rule

For any two events E and F, .)and()()()or( FEPFPEPFEP

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Example 1 : A standard deck of cards contain 52 cards. One card is randomly selected from the deck. Compute the probability of randomly selecting a two or club from a deck of cards.

)cluband2()club()2()clubor2( PPPP

52

4

52

13

52

1

52

16308.0or

13

4

Presented by DW





Presented by DW


Objective C : Complement Rule

Complement Rule

If E represents any event and Ec represents the complement of E, then

)(1)( EPEP C

e.g. The chance of raining tomorrow is 70%. What is the probability that it will not rain tomorrow?

)raining(1)rainingnot( PP

7.01 3.0

Presented by DW


Example 1 : A probability experiment is conducted in which the sample space of the experiment is .

Let event , event , andevent .

}12,11,10,9,8,7,6,5,4,3,2,1{S

(a) List the outcome in E and F. Are E and F mutually exclusive?

(b) Are F and G mutually exclusive? Explain.

Yes, there is no common element between F and G.

}7,6,5,3,2{E }8,7,6,5{F}11,9{G

}7,6,5,3,2{E }8,7,6,5{F

Since there are common elements of 5, 6, and 7 between E and F,E and F are not mutually exclusive.

Presented by DW


(c) List the outcome in E or F. Find by counting the number of outcomes in E or F.

(d) Determine using the General Addition Rule.

}8,7,6,5,3,2{ForE

( )P E or F

6)( ForEN

)or( FEP

12)( SN

)and()()()or( FEPFPEPFEP

12

5

( )P E or F

12

65.0or

2

1

12

4

12

3

12

65.0or

2

1

Presented by DW


(e) List the outcome in EC. Find by counting the number of outcomes in EC.

(f) Determine using Complement Rule.

}12,11,10,9,8,4,1{CE

)(

)()(

SN

ENEP

CC

12

51

( )CP E

( )CP E

( ) 1 ( )CP E P E

12

7

12

7

12

5

12

12

Presented by DW


Example 2: In a large department store, there are 2 managers, 4 department heads, 16 clerks, and 4 stock persons. If a person is selected at random, (a) find the probability that the person is a clerk or a manager; (b) find the probability that the person is not a clerk.

)managerandclerk()manager()clerk()managerorclerk( PPPP

26

16

26

161

)clerk(1)clerkanot( PP

(a)

(b)

26

2

26

0

26

18

13

9

26

16

26

26

26

10

13

5

Presented by DW


Example 3: The following probability show the distribution for the number of rooms in U.S. housing units.

Rooms Probability

One 0.005

Two 0.011

Three 0.088

Four 0.183

Five 0.230

Six 0.204

Seven 0.123

Eight or more 0.156

Source: U.S. Censor Bureau

(a) Verify that this is a probability model.Is each probability outcome between 0 and 1? Yes

1156.0123.0204.0230.0183.0088.0011.0005.0

Is ? )1)(( ieP

YesPresented by DW


(b) What is the probability that a randomly selected housing unit has four or more rooms? Interpret this probability.

156.0123.0204.0230.0183.0)roomsmoreor4( P

896.0

Approximate 89.6% of housing unit has four or more rooms.

Presented by DW

Example 4: According the U.S. Censor Bureau, the probability that a randomly selected household speaks only English at home is 0.81. The probability that a randomly selected household speaks only Spanish at home is 0.12.(a) What is the probability that a randomly selected household speaks only English or only Spanish at home?

(b) What is the probability that a randomly selected household speaks a language other than only English at home?

(c) Can the probability that a randomly selected household speaks only Polish at home equal to 0.08? Why or why not?

)onlySPNandonlyENG(

)onlySPN()onlyENG()onlySPNoronlyENG(

P

PPP

)onlyENG(1)onlyENGnot( PP

81.0

81.01

1)( iePNo, because 81.0i.e. 12.0 08.0 01.1 1

12.0 0 93.0

19.0

Presented by DWLos Angeles Mission College





Presented by DW


Objective D : Contingency TableA Contingency table relates two categories of data. It is also called a two-way table which consists of a row variable and a column variable. Each box inside a table is called a cell.

Example 1 : In a certain geographic region, newspapers are classified as being published daily morning, daily evening and weekly. Some have a comics section and other do not. The

distribution is shown here.

(CY)

(M) (E) (W)

(CN)

6

9

5 7 3 15

Have Comics Section Morning Evening WeeklyYes 2 3 1 No 3 4 2

Presented by DW


)and()()()or( CYMPCYPMPCYMP

)(

)()(

SN

WNWP

If a newspaper is selected at random, find these probabilities.

(a) The newspaper is a weekly publication.

(b) The newspaper is a daily morning publication or has comics .

(c) The newspaper is a weekly or does not have comics .

15

5

)and()()()or( CNWPCNPWPCNWP

15

3

15

35

1

15

6

15

2

5

3

15

9

15

9

15

2 10 2

15 3

Presented by DW





Presented by DW


Section 5.3 Independence and Multiplication RuleObjective A : Independent Events

Two events are independent if the occurrence of event E does not affect the probability of event F.

Example 1 : Determine whether the events E and F are independent or dependent. Justify your answer.

Two events are dependent if the occurrence of event E affects the probability of event F.

(a) E: The battery in your cell phone is dead. F: The battery in your calculator is dead.

(b) E: You are late to class. F: Your car runs out of gas.

Independent

Dependent Presented by DW





Presented by DW


Objective B : Multiplication Rule for Independent Events

If E and F independent events, then . )()()( FPEPFandEP

Example 1 : If 36% of college students are underweight, find the probability that if three college students are selected at random, all will be underweight.

Independent case

)tunderweigh3rdandtunderweigh2ndandtunderweigh1st(P

)tunderweigh3rd()tunderweigh2nd()tunderweigh1st( PPP

36.0 36.0 36.0 047.0

Presented by DW


Example 2 : If 25% of U.S. federal prison inmates are not U.S. citizens,

find the probability that two randomly selected federal prison inmates will be U.S. citizens.

Independent case

)citizenU.S.2ndandcitizenU.S.1st(P

75.0

)citizenU.S.2nd()citizenU.S.1st( PP

75.0 563.0

Presented by DW





Presented by DW


Objective C : At-Least Probabilities

Probabilities involving the phrase “at least” typically use the Complement Rule. The phrase at least means “greater than or equal to.” For example, a person must be at least 17 years old to see an R-rated movie.

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Example 1 : If you make random guesses for four multiple-choice test questions (each with five possible answers), what is the probability of getting at least one correct?

5

4)correctnone(P

)correctnone(1)correctoneleastat( PP

Direct method :

Indirect method :

)correctoneleastat(P

)correctfour()correctthree()correcttwo()correctone( PPPP

What is the opposite of at least one correct? None is correct.

4

5

41

5

44

5

4

5

4

5

4

590.0Presented by DW


Example 2 : For the fiscal year of 2007, the IRS audited 1.77% of individual tax returns with income of $100,000 or more. Suppose this percentage stays the same for the current fiscal year.

(a) Would it be unusual for a return with income of $100,000 or more to be audited?

Yes, 1.77% is unusually low chance of being audited. (In general, probability of less than 5% is considered to be unusual.)

(b) What is the probability that two randomly selected returns with income of $100,000 or more to be audited?

Let A be the return with income of $100,000 or more being audited.

)0177.0()2ndand1st( AAP

0000313.0

)0177.0(

Presented by DW


(d) What is the probability that at least one of the two randomly selected returns with income of $100,000 or more to be audited?

)not2ndandnot1st( AAP

)9823.0(

(c) What is the probability that two randomly selected returns with income of $100,000 or more will NOT be audited?

)none(1)oneleastat( APAP

965.01

)9823.0( 965.0

035.0

Presented by DW

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