probability, population and sample

Basic concepts: Probability, Population & SampleDr. S. A. Rizwan, M.D.

PublicHealthSpecialistSBCM, JointProgram– Riyadh

MinistryofHealth,KingdomofSaudiArabia

Learningobjectives

Demystifying statistics! – Lecture 1 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Familiarisewithtermsusedinprobability• Defineprobability• Describethe3approachesofprobability• Understandthebasiclawsofprobability• Solveproblemsbasedonabovelaws

• DescribetheconceptsofPopulationandSample

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Section1:Probabilitybasics

Demystifying statistics! – Lecture 1 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh 3

Afewimportantterms


• Randomexperiment• Samplespace• Event

• exhaustive,impossible,elementary,composite,certain,mutuallyexclusive,independent,dependent,favourable,equallylikely

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Afewimportantterms


Eventtype ExampleIndependent Twoseedsaresown,germinationofoneis

notaffectedbytheother.

Dependent Ifwedrawacardfromapackofwellshuffledcards,ifthefirstcarddrawnisnotreplacedthentheseconddrawisdependentonthefirstdraw.

Mutuallyexclusive

Inobservationofseedgerminationtheseedmayeithergerminateoritwillnotgerminate.

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Probability– whatisit?


• Chancethatsomethingwillhappen,howlikelyaneventwillhappen

• Canbemeasuredwithanumberlike"10%chanceofrain",orusingwordslikeimpossible,unlikely,possible,evenchance,likelyandcertain

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Probability– whatisit?


• Exampleofafairdie• P(landinga5)• P(landinganevennumber)

• Exampleofmarbles

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Probability– approachestocalculate


• Theoretical probability• Relativefrequency• Subjectiveprobability

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LawsofProbability– Setanalogy


• Union• Intersection• Complement

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Thoughtexercises:


Example1.Abagcontains5redmarbles,3bluemarbles,and2greenmarbles.

Q1.pr (red)+pr (blue)+pr (green)Q2.pr (red)+pr (notred)Q3.pr (redorgreen)

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Thoughtexercises:


Example2.Rolladieandflipacoin.

Q1.pr (headsandrolla3)

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Thoughtexercises:


Example3.Thereare20peopleintheroom:12girls(5withblondhairand7withbrownhair)and8boys(4withblondhairand4withbrownhair).Thereareatotalof9blondsand11withbrownhair.

Q1.pr (girlorblond)Q2.pr (girlwithbrownhair)andpr (girl)

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Thoughtexercises:


Example4.tossacoin4times

Q1.Whatistheprobabilityofgettingatleastoneheadonthe4tossespr (atleastoneH)

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Thoughtexercises:


Example5.Anindividualapplyingtoacollegedeterminesthathehasan80%chanceofbeingaccepted,andheknowsthatdormitoryhousingwillonlybeprovidedfor60%ofacceptedstudents.

Q1.Whatisthechanceofthestudentbeingacceptedandreceivingdormitoryhousing?

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Thoughtexercises:


Example6.A =“Patienthasliverdisease.”Pastdatatellsyouthat10%ofpatientsenteringyourclinichaveliverdisease.B =“Patientisanalcoholic.”Fivepercentoftheclinic’spatientsarealcoholics.Amongpatientsdiagnosedwithliverdisease,7%arealcoholics.

Q1.Findoutapatient’sprobabilityofhavingliverdiseaseiftheyareanalcoholic.

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LawsofProbability- 1


• Theprobabilityofaneventisbetween0and1

• Aprobabilityof1isequivalentto100%certainty

• Probabilitiescanbeexpressedatfractions,decimals,orpercentage

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• Thesumoftheprobabilitiesofallpossibleoutcomesis1or100%.• IfA,B,andCaretheonlypossibleoutcomes,

• thenp(A)+p(B)+p(C)=1

Example: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. p (red) + p (blue) + p (green) = 1

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• Thesumoftheprobabilityofaneventoccurringanditnotoccurringis1.• p(A)+p(notA)=1• p(notA)=1- p(A)

Example: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. p (red) + p (not red) = 1

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• IftwoeventsAandBareindependent• thentheprobabilityofAandBoccurringistheproductoftheirindividualprobabilities.

• p(AandB)=p(A)Xp(B)

Example: roll a die and flip a coin. p (heads and roll a 3) = p (H) X p (3)

*MultiplicativeTheorem

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LawsofProbability- 4a


• IftwoeventsAandBaredependent• thentheprobabilityofAandBoccurringis• p(AandB)=p(B|A)Xp(A)=p(A|B)Xp(B)

• Anindividual applyingtoacollegedeterminesthathehasan80%chanceofbeingaccepted,andheknowsthatdormitoryhousingwillonlybeprovidedfor60%ofacceptedstudents.

• Whatisthechanceofthestudentbeingacceptedandreceivingdormitoryhousing?• P(AcceptedandDormitoryHousing)• =P(DormitoryHousing|Accepted)*P(Accepted)• =(0.60)*(0.80)=0.48

*MultiplicativeTheorem

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• IftwoeventsAandBaremutuallyexclusive• thentheprobabilityofAorBoccurringisthesumoftheirindividualprobabilities.

• p(AorB)=p(A)+p(B)

Example: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. p (red or green) = p (red) + p (green)

*AdditionTheorem

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• IftwoeventsAandBareNOT mutuallyexclusive• thentheprobabilityofAorBoccurringisthesumoftheirindividualprobabilitiesminustheprobabilityofbothAandBoccurring

• p(AorB)=p(A)+p(B)– p(AandB)

Example: 12 girls (5 with blond hair and 7 with brown hair) and 8 boys (4 with blond hair and 4 with brown hair). There are a total of 9 blonds and 11 with brown hair. One person from the group is chosen randomly. p (girl or blond) = p (girl) + p (blond) – p (girl and blond)

*AdditionTheorem

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• Theprobabilityofatleastoneeventoccurringoutofmultipleeventsisequaltooneminustheprobabilityofnoneoftheeventsoccurring.• p(atleastone)=1– p(none)

Example: What is the probability of getting at least one head on the 4 throw of a coin?p (at least one H) = 1 – p (no H) = 1 – p (TTTT)

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• IfeventBisasubsetofeventA,• thentheprobabilityofBislessthanorequaltotheprobabilityofA.• p(B)≤p(A)

Example: There are 20 people in the room: 12 girls (5 with blond hair and 7 with brown hair) and 8 boys (4 with blond hair and 4 with brown hair).p (girl with brown hair) ≤ p (girl)

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ConditionalProbability


• TwoeventsAandBaresaidtobedependent,whenBcanoccuronlywhenAisknowntohaveoccurred(orviceversa)

• TheprobabilityattachedtosuchaneventiscalledtheconditionalprobabilityandisdenotedbyP(A/B)

*BayesTheorem

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ConditionalProbability


*BayesTheorem

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ConditionalProbability- Example


• A =“Patienthasliverdisease.”Pastdatatellsyouthat10%ofpatientsenteringyourclinichaveliverdisease.P(A)=0.10.

• B =“Patientisanalcoholic.”Fivepercentoftheclinic’spatientsarealcoholics.P(B)=0.05.

• Amongpatientsdiagnosedwithliverdisease,7%arealcoholics.Thisisyour B|A.

• Findingoutapatient’sprobabilityofhavingliverdiseaseiftheyareanalcoholic.

• P(A|B)=(0.07*0.1) / 0.05=0.14

*BayesTheorem

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Section2:Population&Sample


Population


• Thepopulationisthesetofentitiesunderstudy• Apopulationincludesalloftheelementsfromasetofdata.• Ameasurablecharacteristicofapopulation- parameter

Sample


• Asampleconsistsofoneormoreobservationsfromthepopulation• Ameasurablecharacteristicofasampleiscalledastatistic.

Sample


Descriptivestatistics


• Descriptivestatisticsprovideaconcisesummaryofdata.• Youcansummarizedatanumericallyorgraphically

Descriptivestatistics


Inferentialstatistics


• Inferentialstatisticsusearandomsampleofdatatakenfromapopulationtodescribeandmakeinferencesaboutthepopulation.

Inferentialstatistics


Takehomemessages


• Understandingprobabilityanditslawsareimportanttounderstandingbiostatistics

• Theconceptsofpopulationandsampleareessentialforunderstandinginferentialstatistics

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probability, population and sample

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