11 Probability

8/8/2019 11 Probability

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STATISTICSSTATISTICS -- II

THE NATURE OFTHE NATURE OF STATSTATIISTSTIICSCS AND PROBABILITYAND PROBABILITY

SAMPLING TECHNIQUESSAMPLING TECHNIQUESFREQUENCY DISTRIBUTIONS AND GRAPHSFREQUENCY DISTRIBUTIONS AND GRAPHS

OTHER TYES OF GRAPHSOTHER TYES OF GRAPHS

DATA DESCRIPTIONDATA DESCRIPTION

COUNTING TECHNIQUESCOUNTING TECHNIQUES

PROBABILITYPROBABILITY

BAYESS THEOREMBAYESS THEOREM

PROBABILITY DISTRIBUTIONSPROBABILITY DISTRIBUTIONS

THE BTHE BIINOMNOMIIAL DAL DIISTRSTRIIBUTBUTIIONON

THE MULTINOMIAL DISTRIBUTIONTHE MULTINOMIAL DISTRIBUTIONTHE POISSON DISTRIBUTIONTHE POISSON DISTRIBUTION

TTHE HYPERGEOMETRHE HYPERGEOMETRIIC DC DIISTRSTRIIBUTBUTIIONON

PPROPERTIES OF THE NORMAL DISTRIBUTIONROPERTIES OF THE NORMAL DISTRIBUTION


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OUTLINEOUTLINESample Spaces And Probability ExperimentsSample Spaces And Probability Experiments

Types of probabilityTypes of probability

ClassicalClassical

EmpricalEmprical

SubjectiveSubjective

SAMPLE SPACES AND PROBABILITYSAMPLE SPACES AND PROBABILITY


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ProbabilityProbability asas aa generalgeneral conceptconcept cancan bebe defineddefined asas thethechancechance ofof anan eventevent occurringoccurring.. MostMost peoplepeople areare familiarfamiliar withwithprobabilityprobability fromfrom observingobserving oror playingplaying gamesgames ofof chance,chance,suchsuch asas cardcard games,games, slotslot machines,machines, oror lotterieslotteries.. InIn additionadditiontoto beingbeing usedused inin gamesgames ofof chance,chance, probabilityprobability theorytheory isisusedused inin thethe fieldsfields ofof insurance,insurance, investments,investments, andand weatherweather

forecasting,forecasting, andand inin variousvarious otherother areasareas.. Finally,Finally, asas statedstated ininChapterChapter 11,, probabilityprobability isis thethe basisbasis ofof inferentialinferential statisticsstatistics..ForFor example,example, predictionspredictions areare basedbased onon probability,probability, andandhypotheseshypotheses areare testedtested byby usingusing probabilityprobability..

TheThe basicbasic conceptsconcepts ofof probabilityprobability areare explainedexplained inin thisthischapterchapter.. TheseThese conceptsconcepts includeinclude probabilityprobability experiments,experiments,samplesample spaces,spaces, thethe additionaddition andand multiplicationmultiplication rules,rules, andandthethe probabilitiesprobabilities ofof complementarycomplementary eventsevents.. SectionSection 55--66explainsexplains howhow thethe countingcounting rulesrules of of ChapterChapter 44 andand thetheprobabilityprobability rulesrules cancan bebe usedused togethertogether toto solvesolve aa widewidevarietyvariety ofof problemsproblems..



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TheThe theorytheory ofof probabilityprobability grewgrew outout of of thethe studystudy ofof variousvariousgamesgames ofof chancechance usingusing coins,coins, dice,dice, andand cardscards.. SinceSince thesethesedevicesdevices lendlend themselvesthemselves wellwell toto thethe applicationapplication ofof conceptsconceptsofof probability,probability, theythey willwill bebe usedused inin thisthis chapterchapter asasexamplesexamples.. ThisThis sectionsection beginsbegins byby explainingexplaining somesome basicbasic

conceptsconcepts ofof probabilityprobability.. ThenThen thethe typestypes ofof probabilityprobability andandprobabilityprobability rulesrules areare discusseddiscussed..

ConseptsConsepts

ProcessesProcesses suchsuch asas flippingflipping aa coin,coin, rolling,rolling, aa die,die, oror drawingdrawing

aa cardcard fromfrom aa deckdeck areare calledcalledprobabilityprobability experimentsexperiments..

A probability experiment is a process that leads to well-definedresults called outcomes.

AnAn outcomeoutcome isis thethe resultresult ofof aa singlesingle trialtrial of of aa probabilityprobabilityexperimentexperiment..



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AA trialtrial meansmeans flippingflipping aa coincoin once,once, rollingrolling oneone diedie once,once, ororthethe likelike.. WhenWhen aa coincoin isis tossed,tossed, therethere areare twotwo possiblepossibleoutcomesoutcomes:: headhead oror tailtail.. ((NoteNote:: WeWe excludeexclude thethe possibilitypossibility ofofaa coincoin landinglanding onon itsits edgeedge..)) InIn thethe rollroll ofof aa singlesingle die,die, therethereareare sixsix possiblepossible outcomesoutcomes:: 11,, 22,, 33,, 44,, 55,, oror 66.. InIn anyanyexperiment,experiment, thethe setset of of allall possiblepossible outcomesoutcomes isis calledcalled thethe

samplesample spacespace..

A samplespaceA samplespaceis the set of all possible outcomes of ais the set of all possible outcomes of aprobability experiment.probability experiment.

Some sample spaces for various probability experiments areSome sample spaces for various probability experiments areshown here.shown here.

Experiment Sample space

Toss one coin

Roll a die

Answer a true-false question

Toss two coins

Head, tail

1,2,3,4,5,6

True, false

Head-head, tail-tail, head-tail, tail-head



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ItIt isis importantimportant toto realizerealize thatthat whenwhen twotwo coinscoins arearetossed,tossed, therethere areare four four possiblepossible outcomes,outcomes, asasshownshown inin thethe fourthfourth experimentexperiment aboveabove.. BothBoth

coinscoins couldcould fallfall headsheads upup.. BothBoth coinscoins couldcould fallfalltailstails upup.. CoinCoin 11 couldcould fallfall headsheads upup andand coincoin 22tailstails upup.. OrOr coincoin 11 couldcould fallfall tailstails upup andand coincoin 22headsheads upup.. HeadsHeads andand tailstails willwill bebe abbreviatedabbreviated asasHH andand TT throughoutthroughout thisthis chapterchapter..



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Example 5-1

Find the sample space for rolling two dice.

Solution Since each die can land in six different ways, and two diceSince each die can land in six different ways, and two dice

are rolled, the sample space can be presented by aare rolled, the sample space can be presented by arectangular array, as shown in Figure 5rectangular array, as shown in Figure 51. The sample1. The sample

space is the list of pairs of numbers in the chart.space is the list of pairs of numbers in the chart.

Die 1Die 1Die 2Die 2

11 22 33 44 55 66

11 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

22 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

33 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

44 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

55 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

66 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Figure5-1Sample Space forRolling Two Dice

(Example 5-1)



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Example 5-2

FindFind thethe samplesample spacespace forfor drawingdrawing oneone cardcard fromfrom ananordinaryordinary deckdeck ofof cardscards..

Solution

Since there are four suits (hearts, clubs, diamonds, andSince there are four suits (hearts, clubs, diamonds, andspades) and 13 cards for each suit (ace through king), therespades) and 13 cards for each suit (ace through king), there

are 52 outcomes in the sample space.are 52 outcomes in the sample space.



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Example5-3

FindFind thethe samplesample spacespace forfor thethe gendergender ofof thethe childrenchildren ifif aafamilyfamily hashas threethree childrenchildren.. UseUse BB forfor boyboy andand GG forfor girlgirl..

Solution

ThereThere areare twotwo genders,genders, malemale andand female,female, andand eacheach childchildcouldcould bebe eithereither gendergender.. Hence,Hence, therethere areare eighteight possibilities,possibilities,asas shownshown herehere..

BBB BBG BGB GBB GGG GGB GBG BGGBBB BBG BGB GBB GGG GGB GBG BGG

InIn thethe previousprevious examples,examples, thethe samplesample spacesspaces werewere foundfoundbyby observationobservation andand reasoningreasoning;; however,however, aa treetree diagramdiagramcancan alsoalso bebe usedused.. TheThe treetree diagramdiagram cancan alsoalso bebe usedused asas aasystematicsystematic wayway toto find find all all possiblepossible outcomesoutcomes ofof aaprobabilityprobability experimentexperiment..



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Example5-4

Use a tree diagram to find the sample space for the genderUse a tree diagram to find the sample space for the genderof three children in a family, as in Example 5of three children in a family, as in Example 5--3.3.

SolutionSolution

There are two possibilities for the first child, two for theThere are two possibilities for the first child, two for thesecond, and two for the third. Hence, the tree diagram cansecond, and two for the third. Hence, the tree diagram canbe drawn as shown in Figure 5be drawn as shown in Figure 5--3.3.

G

B B

B

G

G

G

B G B G B BG

G G G G G B G B G G B B G GB GB B B B G B B B

Figure5Figure5--33Diagram forDiagram for

childrenchildren


1st

2nd

3rd


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AnAn outcomeoutcome waswas defineddefined previouslypreviously asas thethe resultresult ofof aa singlesingle

trialtrial ofof aa probabilityprobability experimentexperiment.. InIn manymany problems,problems, oneonemustmust findfind thethe probabilityprobability ofof twotwo oror moremore outcomesoutcomes.. ForFor thisthisreason,reason, itit isis necessarynecessary toto distinguishdistinguish betweenbetween anan outcomeoutcomeandand anan eventevent..

An event consists of one or more outcomes of a probabilityexperiment.

AnAn eventevent cancan bebe oneone outcomeoutcome oror moremore thanthan oneone outcomeoutcome..ForFor example,example, ifif aa diedie isis rolledrolled andand aa 66 shows,shows, thisthis resultresult isiscalledcalled anan outcome,outcome, sincesince itit isis aa resultresult ofof aa singlesingle trialtrial.. AnAneventevent withwith oneone outcomeoutcome isis calledcalled aa simplesimple eventevent.. TheTheeventevent ofof gettinggetting anan oddodd numbernumber isis calledcalled aa compoundcompound

event,event, sincesince itit consistsconsists ofof threethree outcomesoutcomes oror threethree simplesimpleeventsevents.. InIn general,general, aa compoundcompound eventevent consistsconsists ofof twotwo orormoremore outcomesoutcomes oror simplesimple eventsevents..



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ThereThere areare threethree basicbasic typestypes of ofprobabilityprobability::

1.1. ClassicalClassical probabilityprobability

2.2. EmpiricalEmpirical oror relativerelative frequencyfrequency probabilityprobability

3.3. SubjectiveSubjective probabilityprobability



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ClassicalProbability

ClassicalprobabilityClassicalprobabilityuses sample spaces to determineuses sample spaces to determinethe numerical probability that an event will happen. Onethe numerical probability that an event will happen. Onedoes not actually have to perform the experiment todoes not actually have to perform the experiment todetermine that probability. Classical probability is so nameddetermine that probability. Classical probability is so namedbecause it was the first type of probability studied formallybecause it was the first type of probability studied formallyby mathematicians in the 17by mathematicians in the 17thth and 18and 18thth centuries.centuries.

Classical probability assumes that all outcomes in theClassical probability assumes that all outcomes in thesample space are equally likely to occur.sample space are equally likely to occur. For example,For example,when a single die is rolled, each outcome has the samewhen a single die is rolled, each outcome has the sameprobability of occurring. Since there are six outcomes, eachprobability of occurring. Since there are six outcomes, eachoutcome has a probability ofoutcome has a probability of 1/61/6 . When a card is selected. When a card is selectedfrom an ordinary deck of 52 cards, one assumes that thefrom an ordinary deck of 52 cards, one assumes that thedeck has been shuffled, and each card has the samedeck has been shuffled, and each card has the sameprobability of being selected. In this case, it isprobability of being selected. In this case, it is 1/521/52..

Equallylikelyeventsare events that have the same probabilityof occurring.



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The probability of any eventThe probability of any event EE isis

number of outcomes innumber of outcomes in EE

total number of outcomes in the sample spacetotal number of outcomes in the sample space

This probability is denoted byThis probability is denoted by

This probability is calledThis probability is called classical probability,classical probability, and it uses the sample spaceand it uses the sample space S.S.

( )

( ) ( )

n

P n S

Formula for ClassicalProbabilityFormula for ClassicalProbability

ProbabilitiesProbabilities cancan bebe expressedexpressed asas fractions,fractions, decimals,decimals, ororwherewhere appropriateappropriate percentagespercentages.. IfIf oneone asks,asks, "What"What isis thethe

probabilityprobability ofof gettinggetting aa headhead whenwhen aa coincoin isis tossed?"tossed?" typicaltypicalresponsesresponses cancan bebe anyany ofof thethe followingfollowing threethree..

"One"One--halfhalf..""

"Point"Point fivefive..""

"Fifty

"Fifty percentpercent..

""



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These answers are all equivalent. In most cases,These answers are all equivalent. In most cases,the answers to examples and exercises given inthe answers to examples and exercises given inthis chapter are expressed as fractions orthis chapter are expressed as fractions ordecimals, but percentages are used wheredecimals, but percentages are used whereappropriate.appropriate.

Probabilities should be expressed as reducedProbabilities should be expressed as reducedfractions or rounded to two or three decimalfractions or rounded to two or three decimalplaces. However, when obtaining probabilitiesplaces. However, when obtaining probabilities

from one of the tablesfrom one of the tables given to you,given to you, use theuse thenumber of decimal places given in the table. Ifnumber of decimal places given in the table. Ifdecimals are converted to percentages to expressdecimals are converted to percentages to expressprobabilities, move the point two places to theprobabilities, move the point two places to theright and add a percent sign.right and add a percent sign.



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Example5-6 If a family has three children, find the probability that allIf a family has three children, find the probability that all

the children are girls.the children are girls.

Solution

The sample space for the gender of children for a familyThe sample space for the gender of children for a familythat has three children is BBB, BBG, BGB, GBB, GGG, GGB,that has three children is BBB, BBG, BGB, GBB, GGG, GGB,GBG, and BGG (see Examples 5GBG, and BGG (see Examples 5--3 and 53 and 5--4). Since there is4). Since there isone way in eight possibilities for all three children to beone way in eight possibilities for all three children to begirls,girls,

1(GGG)

8P !



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Example 5-7A cardisdrawn fromanA cardisdrawn froman

ordinarydeckordinarydeck..FindtheseFindthese

probabilities.probabilities.

a.a. Ofgettinga jack.Ofgettinga jack.

b.b. Ofgettingthe6ofclubs.Ofgettingthe6ofclubs.

Solutiona.a. Thereare4Thereare4 jacksand52jacksand52

possiblepossibleoutcomes. Hence,outcomes. Hence,

b.b. Sincethereisonlyone6Sincethereisonlyone6ofclubs,theprobabilityofofclubs,theprobabilityofgettinga6ofclubsisgettinga6ofclubsis

4 1(jack)

52 13P ! !

1(6 ofclubs)

52

P !

SAMPLE SPACES AND PROBABILITYSAMPLE SPACES AND PROBABILITYA

k

2

k

3

k

4

k

5

k

6

k

7

k

8

k

9

k

10

k

J

k

Q

k

K

kA

j

2

j

3

j

4

j

5

j

6

j

7

j

8

j

9

j

10

j

J

j

Q

j

K

jA

l

2

l

3

l

4

l

5

l

6

l

7

l

8

l

9

l

10

l

J

l

Q

l

K

lA

i

2

i

3

i

4

i

5

i

6

i

7

i

8

i

9

i

10

i

J

i

Q

i

K

i


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Example5Example5--88

When a single die is rolled, find the probability ofWhen a single die is rolled, find the probability ofgetting a 9.getting a 9.

SolutionSolution

Since the sample space is 1,2, 3, 4, 5, and 6, it isSince the sample space is 1,2, 3, 4, 5, and 6, it isimpossible to get aimpossible to get a 9. Hence, the probability is9. Hence, the probability is

When the event is certain to occur, the probabilityWhen the event is certain to occur, the probability

is one, as shown in the next example.is one, as shown in the next example.

0(9) 06

P ! !



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Example5-9

When a single die is rolled, what is the probability ofWhen a single die is rolled, what is the probability ofgetting a number less than 7?getting a number less than 7?

Solution

Since all outcomes, 1, 2, 3, 4, 5, and 6, are less than 7, theSince all outcomes, 1, 2, 3, 4, 5, and 6, are less than 7, theprobability isprobability is

The event of getting a number less than 7 is certain.The event of getting a number less than 7 is certain.

6

(numberless than 7) = 16P!



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Finally, the sum of the probabilities of all outcomes in aFinally, the sum of the probabilities of all outcomes in asample space is one.sample space is one.

For example, in the role of a fair die, each outcome in theFor example, in the role of a fair die, each outcome in the

sample space has a probability ofsample space has a probability of 1/61/6. Hence, the sum of. Hence, the sum of

the probabilities of the outcomes is as shown.the probabilities of the outcomes is as shown.

+++++Sum

Probability

654321Outcome

1

6

1

6

1

6

1

6

1

6

1

6

1

6

1

6

1

6

1

6

1

6

1 61

6 6! !



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ComplementaryEventsComplementaryEvents

Another important concept in probability theory is that ofAnother important concept in probability theory is that ofcomplementary events.complementary events. When a die is rolled, for instance,When a die is rolled, for instance,the sample space consists of the outcomes of 1, 2, 3, 4, 5,the sample space consists of the outcomes of 1, 2, 3, 4, 5,and 6. The eventand 6. The event EEof getting odd numbers consists of theof getting odd numbers consists of the

outcomes of 1, 3, and 5. The event of not getting an oddoutcomes of 1, 3, and 5. The event of not getting an oddnumber is called thenumber is called the complementcomplementof eventof event EE,, and it consistsand it consistsof the outcomes 2, 4, and 6.of the outcomes 2, 4, and 6.

The complementofaneventf is the set of outcomes in thesample space that are not included in the outcomes of event E.The complement ofE is denoted by (read "E bar").

The next example further illustrates the concept ofThe next example further illustrates the concept ofcomplementary events.complementary events.



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The outcomes of an event and the outcomes of theThe outcomes of an event and the outcomes of thecomplement make up the entire sample space. For example,complement make up the entire sample space. For example,if two coins are tossed, the sample space is HH, HT, TH, andif two coins are tossed, the sample space is HH, HT, TH, andTT. The complement of "getting all heads" is not "getting allTT. The complement of "getting all heads" is not "getting alltails," since the event "all heads" is HH, and the complementtails," since the event "all heads" is HH, and the complementof HH is HT, TH, and TT. Hence, the complement of theof HH is HT, TH, and TT. Hence, the complement of the

event "all heads" is the event "getting at least one tail.event "all heads" is the event "getting at least one tail.

Since the event and its complement make up the entireSince the event and its complement make up the entiresample space, it follows that the sum of the probability ofsample space, it follows that the sum of the probability ofthe event and the probability of its complement will equal 1.the event and the probability of its complement will equal 1.

That is,That is, P(E) + P(P(E) + P() = 1) = 1. In the previous example, let. In the previous example, letEE= all= allheads, or HH, and letheads, or HH, and let = at least one tail, or HT, TH, TT= at least one tail, or HT, TH, TT

ThenThen ;; hence,hence, 1 3( ) ( ) 14 4

PE PE ! !1 3

( ) ( )4 4

PE PE! !and



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Rule for Complementary EventsRule for Complementary Events

( ) 1 ( ) ( ) 1 ( ) ( ) ( ) 1PE PE PE PE PE PE! ! !or or

Stated in words, the rule is: If the probability of an eventor the probability of its complement is known, then theother can be found by subtracting the probability from 1.

This rule is important in probability theory because at timesthe best solution to a problem is to find the probability ofthe complement of an event and then subtract from 1 toget the probability of the event itself.


The rule for complementary events can be statedThe rule for complementary events can be statedalgebraically in three ways.algebraically in three ways.


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ExampleExample55--1111

If the probability that a person lives in an industrializedIf the probability that a person lives in an industrialized

country of the world iscountry of the world is , find the probability that a, find the probability that a

person does not live in an industrialized country.person does not live in an industrialized country.

Source:Source: Harper's IndexHarper's Index289, no. 1737 (February 1995), p. 11.289, no. 1737 (February 1995), p. 11.

SolutionSolution

PP (not living in an industrialized country)(not living in an industrialized country) =11PP (living in(living in

an industrializedan industrialized country)country)

1

5

1 41

5 5

! !



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Probabilities can be represented pictorially byProbabilities can be represented pictorially by VennVenndiagrams.diagrams. Figure 5Figure 5--4(a) shows the probability of a simple4(a) shows the probability of a simpleeventevent E.E. The area inside the circle represents theThe area inside the circle represents theprobability of eventprobability of event EEthat is,that is, PP((E)E).. The area inside theThe area inside therectangle represents the probability of all the events in therectangle represents the probability of all the events in thesample space,sample space, P(S)P(S)..

The Venn diagram that represents the probability of theThe Venn diagram that represents the probability of thecomplement of an eventcomplement of an event P(P()) is shown in Figure 5is shown in Figure 5--4(b). In4(b). Inthis case,this case, P(P()) 11P(E)P(E),, which is the areawhich is the area inside theinside therectangle but outside the circle representingrectangle but outside the circle representing P(E)P(E).. RecallRecall

thatthat P(S) =P(S) = 11 andand PP((E)E) = 1= 1

P(P().). The reasoning is thatThe reasoning is that PP((E)E) isisrepresented by the area of the circle andrepresented by the area of the circle and P(P()) is theis theprobability of the events that are outside the circle.probability of the events that are outside the circle.



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Figure5-4

Venn Diagram for the Probability and ComplementVenn Diagram for the Probability and Complement

P(S)=1 P()

P(E) P(E)

(a) Simple probability (b) P() = 1 - P(E)



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EmpiricalProbability

The difference between classical andThe difference between classical and empiricalprobabilityempiricalprobabilityis that classical probability assumes that certain outcomesis that classical probability assumes that certain outcomesare equally likely (such as the outcomes when a die isare equally likely (such as the outcomes when a die isrolled) while empirical probability relies on actual experiencerolled) while empirical probability relies on actual experience

to determine the likelihood of outcomes.to determine the likelihood of outcomes.

In empirical probability, one might actually roll a given dieIn empirical probability, one might actually roll a given die66000000times and observe the relative frequencies and usetimes and observe the relative frequencies and usethese frequencies to determine the probability of anthese frequencies to determine the probability of anoutcomeoutcome



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Suppose, forexample,thataresearcherasked25peopleSuppose, forexample,thataresearcherasked25peopleiftheylikedthetasteofanewsoftdrink.Theresponsesiftheylikedthetasteofanewsoftdrink.Theresponseswereclassifiedas "yes," "no," or "undecided." Theresultswereclassifiedas "yes," "no," or "undecided." Theresultswerecategorizedina frequencydistribution,asshown.werecategorizedina frequencydistribution,asshown.

ResponseResponse FrequencyFrequency

YesYes 1515

NoNo 88

UndecidedUndecided 22

TotalTotal 2525


Probabilities now can be compared for various categories.Probabilities now can be compared for various categories.

For example, the probability of selecting a person who likedFor example, the probability of selecting a person who liked

the taste isthe taste is , or, or , since 15 out of 25 people in the, since 15 out of 25 people in the

survey answered "yes."survey answered "yes."

15

25

3

5


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Formula forEmpiricalProbability

Given a frequency distribution,Given a frequency distribution, the probability of an eventthe probability of an eventbeing in a given class isbeing in a given class is

This probability is calledThis probability is called empirical probabilityempirical probabilityand is basedand is basedon observation.on observation.

( ) =f

PEn

!frequency for the class

total frequenciesin the distribution



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In the softIn the soft--drink survey just described, find the probabilitydrink survey just described, find the probability

that a person responded "no.that a person responded "no.

SolutionSolution

8( ) =25

fPEn

!



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In a sample of 50 people, 21 had type O blood, 22 had typeIn a sample of 50 people, 21 had type O blood, 22 had typeA blood, 5 had type B blood, and 2 had type AB blood. SetA blood, 5 had type B blood, and 2 had type AB blood. Setup a frequency distribution and find the followingup a frequency distribution and find the followingprobabilities:probabilities:

aa.. A person has type O blood.A person has type O blood.b.b. AA person has type A or type B blood.person has type A or type B blood.

c.c. AA person has neither type A nor type O blood.person has neither type A nor type O blood.

d.d. AA person does not have type AB blood.person does not have type AB blood.

Source: Based on American Red Cross figures presented inSource: Based on American Red Cross figures presented in TheTheBook ofOddsBook ofOdds by Michael D. Shook and Robert L. Shook (Plume, aby Michael D. Shook and Robert L. Shook (Plume, adivision of Penguin Books, 1991).division of Penguin Books, 1991).



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TypeType FrequencyFrequency

AA 2222

BB 55

ABAB 22

OO 2121TotalTotal 5050

21. ( )

50

22 2 27. ( )50 50 50

5 2 7. ( )

20

50

50

. (

fa P

n

b P

c P

d P

! !

! !

! !

Add t fr ncies f t e tw cl sses.

(Neit er A n r O me ns t at a person aseit er type B or typeAB blood.)

O

Aor B

( )

neit er A nor O

no2 48 24

) 1-50 50 25

! ! !

(Find t e probabilityof notAB bysubtracting t e probabilityof typeAB from 1.)

tAB

A personhastypeOblood.A personhastypeOblood.

AApersonhastype A ortypeBblood.personhastype A ortypeBblood.

AApersonhasneithertype Apersonhasneithertype A

nortypeOblood.nortypeOblood.

persondoesnothavepersondoesnothavetype ABblood.type ABblood.

SOLUTION


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Example5-14

Hospital records indicated that maternity patients stayed inHospital records indicated that maternity patients stayed inthe hospital for the number of days shown in thethe hospital for the number of days shown in thedistribution.distribution.

Find these probabilities.Find these probabilities.

a.a. AA patient stayed exactly 5 days.patient stayed exactly 5 days. c. Ac. A patient stayed at most 4 days.patient stayed at most 4 days.

b.b. AA patient stayed less than 6 days.patient stayed less than 6 days. d. Ad. A patient stayed at least 5 days.patient stayed at least 5 days.

Number of days stayedNumber of days stayed FrequencyFrequency33 151544 323255 565666 191977 55

127

127



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Solution

56. (5)

127

15 32 56 103

. ( ) 127 127 127 127

15 32 47. ( )

127 127 127

. (

a P

b P

c P

d P

!

! !

! !

less t an 6 days

(Less t an 6 days meanseit er 3, or 4, or 5 days.)

at most 4 days

(at most, 4 days means 3 or 4 days.)

atle15 56 5 80

)127 127 127 127

! !ast 5 days

(Atleast, 5 days meanseit er 5, or 6, or 7 days.)

SAMPLE SPACES AND PROBABILITYSAMPLE SPACES AND PROBABILITYNumber of days stayedNumber of days stayed FrequencyFrequency

33 151544 323255 565666 191977 55

127127

A patientstayedexactly5days.A patientstayedexactly5days.

AApatientstayedpatientstayed

lessthan6dayslessthan6days


atmost4daysatmost4days


atleast5daysatleast5days


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Empirical probabilities can also be found using a relativeEmpirical probabilities can also be found using a relative

frequency distribution, as shown in Section 2frequency distribution, as shown in Section 2--2.2.

For example, the relative frequency distribution for ExampleFor example, the relative frequency distribution for Example55--12 is12 is

ResponseResponse FrequencyFrequency RelativeRelativefrequencyfrequency

YesYes 1515 0.600.60

NoNo 88 0.320.32

UndecidedUndecided 22 0.080.08

2525 1.001.00

Hence, the probability that a person responded "no" is 0.32, which is

equal to .8

25



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It should be pointed out that the probabilities which theIt should be pointed out that the probabilities which theproportions steadily approach may or may not agree withproportions steadily approach may or may not agree withthose theorized in the classical model. If not, it can havethose theorized in the classical model. If not, it can haveimportant implications, such as "the die is not fair." Pitimportant implications, such as "the die is not fair." Pitbosses in Las Vegas watch for empirical trends that do notbosses in Las Vegas watch for empirical trends that do notagree with classical theories, and they will sometimes takeagree with classical theories, and they will sometimes take

a set of dice out of play if observed frequencies are too fara set of dice out of play if observed frequencies are too far

out of line with classical expected frequencies.out of line with classical expected frequencies.

SubjectiveProbability

The third type of probability is calledThe third type of probability is called subjective probability.subjective probability.SubjectiveprobabilitySubjectiveprobabilityuses a probability value based onuses a probability value based onan educated guess or estimate, employing opinions andan educated guess or estimate, employing opinions andinexact information.inexact information.



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In subjective probability, a person or group makes anIn subjective probability, a person or group makes aneducated guess at the chance that an event will occur. Thiseducated guess at the chance that an event will occur. Thisguess is based on the person's experience and evaluation ofguess is based on the person's experience and evaluation ofa solution. For example, a sportswriter may say that therea solution. For example, a sportswriter may say that thereis a 70% probability that the Pirates will win the pennantis a 70% probability that the Pirates will win the pennantnext year. A physician might say that on the basis of hernext year. A physician might say that on the basis of her

diagnosis, there is a 30% chance the patient will need andiagnosis, there is a 30% chance the patient will need anoperation. A seismologist might say there is an 80%operation. A seismologist might say there is an 80%probability that an earthquake will occur in a certain area.probability that an earthquake will occur in a certain area.These are only a few examples of how subjectiveThese are only a few examples of how subjectiveprobability is used in everyday life.probability is used in everyday life.

All three types of probability (classical, empirical, andAll three types of probability (classical, empirical, andsubjective) are used to solve a variety of problems insubjective) are used to solve a variety of problems in

business, engineering, and other fieldsbusiness, engineering, and other fields..

SPACES AND SAMPLE PROBABILITYSPACES AND SAMPLE PROBABILITY

Documents

11 Probability