Basic Concepts of Probability_updated_Tm_31_5_2.ppt

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1. Chapter 9
Basic Concepts of Probability

2. Learning Outcomes
Whenyouwillcompletethischapter,youwouldbe
ableto-
Basictermsandcontentofprobability
Calculateprobabilitybetweentwoevents
Estimateandinterpretdifferenttypesofrulesofprobability
Identifytherelationanddifferenttypesofprobability
Identifytherelationanddifferenttypesofprobabilitywith
practicalexamples

3. Lecture Contents
BasicTerminologyofProbability
DifferentApproachesofProbability
ApplicationandSignificanceofProbability
LawsofProbability
ProbabilityRules
Joint,MarginalandConditionalProbability
VariousMathematicalProblemsofProbability

4. Probability:Avaluebetweenzeroandone,inclusive,describingtherelativepossibility(chanceor
likelihood)aneventwilloccur.
Experiment:Experimentisanactthatcanberepeatedundergivenconditions.
Trial:Unitofanexperimentisknownastrial.Thismeansthattrialisaspecialcaseofexperiment.
Experimentmaybeatrialortwoormoretrials.
Outcomes:Theresultofanexperimentisknownasoutcomes.
Example:Throwingadieisatrialandgetting1or2or3or4or5or6isanoutcome.
EquallylikelyOutcomes:Outcomesofatrialaresaidtobeequallylikelyifwehavenoreasonto
expectanyoneratherthantheother.Example-1)Intossingafaircoin,theoutcomesheadandtailare
equallylikely,2)Inthrowingabalanceddieallthesixfacesareequallylikely.
Basic Terminology of
Probability

5. Basic Terminology of
Probability
MutuallyExclusiveOutcomes:Outcomesorcasesaresaidtobemutuallyexclusiveifthehappening
ofanyoneofthemprecludesthehappeningofallothers.Example-1)Intossingacoin,theoutcomes
hearandtailaremutuallyexclusive.2)Inthrowingadie,thesixoutcomeswhicharethedifferentpoints
onthefacesofthedieismutuallyexclusive.
Exhaustiveoutcomes:Outcomesofanexperimentaresaidtobeexhaustiveiftheyincludeallpossible
outcomes.Example-inthrowingadieexhaustivenumberofoutcomesare6.
Samplespace:Thecollectionofallpossibleoutcomesofarandomexperimentiscalledsamplespace.
SamplespaceisusuallydenotedbySorΩ.Example:1)Ifwetossacoin,thesamplespaceis,Ω={H,T}.
WhereHandTdenotethehearandtailofthecoin,respectively,2)Ifasix–sideddieisthrown,the
samplespaceis,Ω={1,2,3,4,5,6}.

6. Probability

7. Probability

8. Probability

9. Application and
Significance of Probability
Probabilityisthebackboneofstatistics.Ithasvastapplicationineveryfieldof
statistics.Thevariouspracticalapplicationsoftheprobabilityare:
a)Instatisticalinference,theproblemofestimationandthetestofhypothesisare
basedonprobabilitytheory.
b)Thevarioustestofsignificance,viz-Z-test,Chi-squaretest,F-test,t-testare
derivedfromthetheoryofprobability.
c)Decisiontheoriesarebasedonfundamentallawsofprobabilityandexpected
value.
d)Theuseofprobabilitytheoryisincreasingineconomicdecisionmaking.
e)Theuseofsubjectiveprobabilitiesismadewhenactualmeasurementisnot
possible.

10. Laws of Probability

11. Addition Rules
AdditiveLawofProbability:Therearetworulesofaddition,thespecialruleofadditionandthe
generalruleofaddition.
SpecialRuleofAddition:Toapplythespecialruleofaddition,theeventsmustbemutuallyexclusive.If
twoeventsAandBaremutuallyexclusive,thespecialruleofadditionstatesthattheprobabilityofoneor
theotherevent’soccurringequalsthesumoftheirprobabilities.Thisruleisexpressedinthefollowing
formula:
P(A or B)= P(A U B)= P(A) + P(B)
GeneralRuleofAddition:Whentwoeventsbothoccur,theprobabilityiscalledajointprobability.Inthis
situation,weusethegeneralruleofaddition.IfAandBaretwoeventsin,then
P(A or B)= P(A U B)= P(A) + P(B) –P(A∩B)

12. Example:Mr.Alifeelsthattheprobabilitythathewillpassmathematicsis2/3andstatisticsis5/6.Ifthe
probabilitythathewillpassboththecourseis3/5,whatistheprobabilitythathewillpassatleastoneof
thecourse?
Solution:LetMandSbetheeventsthathewillpassthecoursemathematicsandstatistics,respectively.
TheeventMUSmeansthatatleastoneofMorSoccurs.Therefore,accordingtotheadditionrulewe
get,
P(A or B)= P(A U B)= P(A) + P(B) –P(A∩B)
= 2/3 + 5/6 –3/5
= 9/10
Addition Rules

13. Addition Rules
Example:
Inasampleof500students,320saidtheyhadastereo,175saidtheyhadaTV,and100saidtheyhad
both.5saidtheyhadneither.Ifastudentisselectedatrandom,whatistheprobabilitythatthestudent
hasonlyastereoorTV?WhatistheprobabilitythatthestudenthasbothastereoandTV?
Solution:LetSandTbetheeventsthatstudentshadstereoandTV,respectively.Thentheprobability
thatstudenthasonlystereoorTVis-
P (S or T) = P(S) + P (T) –P (S and T)
= 320/500 + 175/500 –100/500
= 0.79
TheprobabilitythatthestudenthasbothastereoandTV:
P (S and TV) = 100/500
= 0.20

14. Joint Probability
Jointprobabilityistheprobabilityoftwoeventsinconjunction.Thatis,itistheprobabilityofbothevents
together.ThejointprobabilityofAandBiswrittenP(A∩B),P(AB)orP(A,B).
Example:Thequestion,"DoyoulikewatchingTV?"wasaskedof100people.Resultsareshowninthe
table.WhatistheprobabilityofarandomlyselectedindividualbeingamalewholikeswatchingTV?
Yes No Total
Male 19 41 60
Female12 28 40
Total31 69 100
Solution:Thisisjustajointprobability.Thenumberof"MaleandlikewatchingTV"dividedbythetotal=
19/100=0.19

15. Marginal Probability
MarginalprobabilityistheprobabilityofA,regardlessofwhethereventBdidordidnotoccur.IfBcanbe
thoughtofastheeventofarandomvariableXhavingagivenoutcome,themarginalprobabilityofAcanbe
obtainedbysummingthejointprobabilitiesoveralloutcomesforX.
Example:Thequestion,"DoyoulikewatchingTV?"wasaskedof100people.Resultsareshowninthe
table.WhatistheprobabilityofarandomlyselectedindividuallikewatchingTV?
Solution:Sincenomentionismadeofgender,thisisamarginalprobability,thetotalwholikewatchingTV
dividedbythetotal=31/100=0.31
Yes No Total
Male 19 41 60
Female12 28 40
Total31 69 100

16. Conditional Probability

17. Example:Thequestion,"DoyoulikewatchingTV?"wasaskedof100people.Resultsareshowninthe
table.Whatistheprobabilityofarandomlyselectedindividualisamaleifitisgiventhathelikeswatching
TV?
Yes No Total
Male 19 41 60
Female12 28 40
Total31 69 100
Solution: The conditional probability M given Y is-
Conditional Probability

19. Conditional Probability

20. Multiplication Rule

21. Multiplication Rule

23. Complement Rule

25. Chapter Exercise
1. The following table gives a two-way classification of all basketball players at a state university who began
their college careers between 2001 and 2005, based on gender and whether or not they graduated.
Graduat
ed
Did Not
Graduat
e
Total
Male 126 55 181
Female133 32 165
Total259 87 346
If one of these players is selected at random, find the following probabilities.
a. P (female or did not graduate)
b. P (graduated and male)
c. P (Graduated | female)

26. 1.Inaclassof120students,60arestudyingEnglish,50arestudyingFrenchand20arestudyingbothEnglish
andFrench.Ifastudentisselectedatrandomfromthisclass,whatistheprobabilitythatheisstudying
EnglishifitisgiventhatheisstudyingFrench?
2.Jackcanhitatarget5outof7chances.Andycanhitatarget6outof11.Findthechancesthatthetargetis
hitoncetheybothtry?
3.Russellisplayinginacricketmatchandagameoffootballattheweekend.Theprobabilitythathisteam
willwinthecricketmatchis0.7,andtheprobabilityofwinningis0.9inthefootball.Whatistheprobabilitythat
histeamwillwininbothmatches?Whatistheprobabilitythathisteamwillnotwininbothmatches?
4.Supposeyouhaveaboxwith3bluemarbles,2redmarbles,and4yellowmarbles.Youaregoingtopull
outonemarble,recorditscolor,putitbackintheboxanddrawanothermarble.Findtheprobabilitythatthe
firstmarbleisredandsecondmarbleisyellow.
5.TheprobabilitythatMr.Xwilldieinthenext20yearsis1/5andtheprobabilitythatMr.Ywilldieinthenext
20yearsis1/7.Whatistheprobabilitythat
i.Bothwilldieinthenext20years.ii.Atleastonewilldieinthenext20years.iii.Neitherwilldieinthenext
20years.
Chapter Exercise

27. Thank You