Lecture (Introduction To Probability).pdf

It is very good for learning probability

1. Introduction To Statistics,
Statistics And Probability
1
Dr.Shabbir Ahmad
Assistant Professor,
Department of Mathematics,
COMSATS University
Islamabad, Wah Campus
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 12/6/2021 8:59:34 AM
Lecture No.

2. Introduction To Probability
Basic Concepts
2
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 12/6/2021 8:59:36 AM

3. In this lecture
•Set Theory and Algebra of Sets
•Basics of Probability
•Random and Non-random Experiment
•Event and Sample Space
•Simple and Compound Events
•Mutually Exclusive Events
•Exhaustive Events, Equally Likely Events
•Rules of Counting
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 12/6/2021 8:59:36 AM

4. Set Theory:
❑Asetisanywell-definedcollectionorlistofdistinctobjects,e.g.agroupof
students,thebooksinalibrary,theintegersbetween1to100,allhumanbeings
onearth,etc.
❑Thetermwell-definedheremeansthatanyobjectmustbeclassifiedaseither
belongingornotbelongingtothesetunderstudy.
❑Theobjectsthatareinaset,arecalledmembersorelementsoftheset.
❑Setsareusuallydenotedbycapitalletterssuchas:�,�,??????,??????,etc.
❑Theelementsofthesetsarerepresentedbysmallletters,suchas:�,�,�,�,??????,etc.
❑Elementsareenclosedbybracestorepresentaset,e.g.�isthesetofvowels,i.e.,
�=�,�,??????,�,�
4
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 12/6/2021 8:59:36 AM

5. Algebra of Sets:
5

6. 6
❑Aplannedactivityorprocesswhoseresultsyieldasetofdata(called
outcomes),iscalledandexperiment.
❑Anoutcomeistheresultofasingletrialofaprobabilityexperiment.
❑Aneventconsistsanycollectionofresultsoroutcomesofaprocedure.
❑Asimpleeventisanoutcomeoraneventthatcannotbefurtherbrokendown
intosimplercomponents.
❑Asamplespace(S)isthesetofallpossibleoutcomesorsimpleeventsofa
probabilityexperiment.
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 12/6/2021 8:59:36 AM

7. Random and Non-random Experiment:
❑Anexperimentwhichproducesdifferentresultseventhoughitis
repeatedalargenumberoftimesunderessentiallysimilarconditions,
iscalledarandomexperiment.
❑Forexample:thetossingofafaircoin,throwingofbalanceddie,
selectingasample,etc.
❑Anexperimentwhichproducesfixresultseventhoughitisrepeateda
largenumberoftimes,iscalledanonrandomexperiment.
❑Forexample:mixtureoftwohydrogenandoneoxygenelementmake
water,Areaofcircle,etc.
7
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 12/6/2021 8:59:36 AM

8. Event and Sample Space
8
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 12/6/2021 8:59:36 AM

9. Sample Space Example 1:
Experiment
Possible outcomes
HeadorTail
Thusthesamplespaceis:
�={??????���,��????????????}
Toss a Coin
9
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 12/6/2021 8:59:36 AM

10. Sample Space Example 2:
Experiment
Possible outcomes
1,2,3,4,5,6
Thusthesamplespaceis:
�={1,2,3,4,5,6}
Roll a Dice
10
Dr. Shabbir Ahmad
Assistant Professor, Department of Mathematics, COMSATS University Islamabad, Wah Campus
Cell # 0323-5332733, 0332-5332733. Date: 12/6/2021 8:59:36 AM

11. Events
❑Aneventisanindividualoutcomeoranynumberofoutcomes(samplepoints)
ofarandomexperimentoratrial
❑Theeventsareclassifiedas:
•SimpleEvents
•CompoundEvents
•MutuallyExclusiveEvents
•ExhaustiveEvents
•EquallyLikelyEvents
In set terminology, any subset of a sample space Sof the experiment, is called
an event.
11

12. Simple and Compound Events:
❑Aneventthatcontainsexactlyonesamplepoint,isdefinedasimpleevent.
❑Acompoundeventcontainsmorethanonesamplepointsandisproducedbythe
unionofsimpleevents.
Forexample:
❑Theoccurrenceofsixwhenadieisthrown,isasimpleevent,
❑whiletheoccurrenceofasumof10withapairofdice,isacompoundevent,asit
canbedecomposedintothreesimpleevents(4,6),(5,5),���(6,4).
12

13. 13
When two dice are rolled

14. Example 3: Simple Events and Sample Spaces:
Inthefollowingtable,weuse“b”todenoteababyboyand“g”todenoteababy
girl.
1birth:Theresultof1girlisasimpleeventandsoistheresultof1boy.
3births:Theresultof2girlsfollowedbyaboy(ggb)isasimpleevent.
3births:Theeventof“2girlsand1boy”isnotasimpleeventbecauseitcanoccurwith
thesedifferentsimpleevents:ggb,gbg,bgg.
14

15. Mutually Exclusive Events
❑TwoeventsA&Bofasingleexperimentaresaidtobemutually
exclusiveordisjointifandonlyiftheycannotbothoccuratsame
time.Forexample,
❑Whenwetossacoin,wegeteitheraheadoratail,butnot
both,thetwoeventsheadandtailarethereforemutually
exclusive.
❑Whenadieisrolled,theoutcomearemutuallyexclusiveaswe
getoneandonlyoneofsixpossibleoutcomes1,2,3,4,5,6.
15

16. Exhaustive Events:
❑Eventsaresaidtobecollectivelyexhaustive,whentheunionofmutually
exclusiveeventsistheentiresamplespaceS.
❑Thusinourcoin-tossingexperiment,headandtailarecollectively
exhaustivesetofevents.
16

17. Equally Likely Events:
❑TwoeventsA&Baresaidtobeequallylikely,whenoneeventisaslikelytooccurasthe
other.
❑Inotherwords,eacheventshouldoccurinequalnumberinrepeatedtrials.
❑Forexample,whenafaircoinistossed,theheadisaslikelytoappearastail,andthe
proportionoftimeseachsideisexpectedtoappearis
1
2
.
❑Similarly,whenadieisrolled,eachpossibleoutcomei.e.,1,2,3,4,5,6areequalchanceto
appear,andtheproportionoftimeseachsideisexpectedtoappearis
1
6
.
17

18. Counting Sample Points:
•ThesamplespaceS,isthesetofallpossibleoutcomesofastatisticalexperiment.
•Eachoutcomeinasamplespaceiscalledasamplepoint.Itisalsocalledanelementoramemberofthe
samplespace.
FundamentalPrincipleofCounting:
•WhenthenumberofsamplepointsinasamplespaceSisverylarge,itbecomesverydifficulttolist
themallandtocountthesamplepoints.
•Insuchcases,aprobabilityproblemmaybesolvedbycountingsamplepointsinS,withoutactually
listingeachelement.
18

19. Counting Sample Points:
Thebasicrulesormethodwhichhelpsustocountthenumberofsamplepointswithout
actuallylistingthemallare:
1.RuleofMultiplication
2.RuleofPermutation
3.RuleofCombination
19

20. 1.Rule of Multiplication:
•Ifacompoundexperimentconsistsoftwoexperimentsuchthat,thefirsthasexactly�
1distinct
outcomesandcorrespondingtoeachoutcomeofthefirsttherecanbe�
2distinctoutcomesofthe
secondexperiment,thenthecompoundexperimenthasexactly�
1�
2outcomes.
•Ingeneraliftherearekexperiments,thentotalsamplepointsofcompoundexperimentare:
�
1�
2�
3×⋯×�
??????
20

21. Example 1:
Statement:Howmanysamplepointsarethereiftwocoinsaretossedtogether?
Solution:Totalno.ofoutcomesthefirstcoinhas=�
1=2,andTotalno.ofoutcomes
thesecondcoinhas=�
2=2,
Therefore,apairofcoinscanlendin:
�
1�
2=2∗2=4��??????���
Thesamplespacewillbeas:
�=????????????,??????�,�??????,��
21

22. Example 2:
Statement:Howmanysamplepointsarethereifthecompoundexperimentconsistoftossingacoin
andthrowingadietogether?
Solution:Totalpossibleoutcomesthefirstexperiment(coin)has=�
1=2,andTotalpossibleoutcomes
thesecondexperiment(die)has=�
2=6,
Totalsamplepoints:�
1�
2=2×6=12points
22

23. Example 3:
Statement:Howmanysamplepointsareinthesamplespacewhenapairofdicearethrowntogether?
Solution:thefirstdiecanlendin�
1=6waysandcorrespondingtoeachtheseconddiecanalsolendin
�
2=6ways,thereforeapairofdicecanlendin:
Thus, Sample Space Points are:
�
1�
2=6×6=36points
23

24. Example 4:
ThreeCoins:Howmanysamplepointsarethereifthreecoinsaretossedatonce?
Solution:Eachofthreecoinscanlendin2possibleways,i.e.HeadandTail,Thus,
�
1=�
2=�
3=2
Thepossiblesituationsare:
ThusSampleSpacePoints:�
1�
2�
3=2×2×2=8points
One Coin
Two
Coins
24

25. Example 4 (Cont.,)
25
One Coin 2
nd
Coin 3
rd
Coin Combined Outcome
H
T
H
T
H
T
H
T
H
T
H
T
H
T
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT

26. Example 5:
TwoCoinsandOneDie:
Solution:Eachoftwocoinshasexactlytwopossibleoutcomes,i.e.�
1=�
2=2,and
onediehasexactly6possibleoutcomes,i.e.�
3=6,thusthepossiblesituationsare:
ThuspossibleSamplePoints:�
1�
2�
3=2×2×6=24points
Two Coins
One Die
26

27. Practice Questions:
1.Howmanysamplepointsareinthesamplespace,whenapairofdiceandone
coinarethrowntogether?
2.Howmanysamplepointsareinthesamplespacewhenthreedicearethrown
together?
3.Howmanysamplepointsareinthesamplespacewhenfourcoinsarethrown
together?
4.Ifanexperimentconsistsofthrowingadieandthendrawingaletteratrandom
fromtheEnglishalphabet,howmanypointsarethereinthesamplespace?
27

28. 2. Rule of Permutation:
•Apermutationisanyorderedsubsetof�objectsfromasetof�distinctobjects.
OR
•Apermutationisanarrangementofallorpartofasetofobjects.
•Thenumberofpermutationsof�objectsis�!(�����������??????�??????).
•Thenumberofpermutationsof�distinctobjectstaken�atatimeisdenotedbysymbol
??????
??????
??????,is
definedas:
??????
??????
??????=
??????!
??????−??????!
, �≤�
Where�!=�(�−1)(�−2)…3×2×1,itisrelevanttonotethat1!=1andthatwedefine0!=
1.
28

29. 3. Rule of Combination:
•Acombinationisanysubsetof�objects,selectingwithoutregardingtheirorder,fromasetof�
distinctobjects.
•Thenumberofcombinationsof�distinctobjectstaken�atatimeisdenotedbysymbol
??????
??????
??????or
??????
??????
(read“���”or“�������"),isdefinedas:
??????
??????
??????=
??????!
??????!??????−??????!
Where�≤�.
29

30. Example 1:
Inoneyear,threeawards(research,teaching,andservice)willbegiventoaclassof25graduatestudents
inastatisticsdepartment.Ifeachstudentcanreceiveatmostoneaward,howmanypossibleselections
arethere?
Solution:Given�=25,�=3,
Sincetheawardsaredistinguishable,itisapermutationproblem.Thetotalnumberofsamplepointsis:
??????
??????
??????=
�!
�−�!
=
25!
25−3!
=
22!×23×24×25
22!
=(23)2425=13,800
30

31. Example 2:
Ayoungboyaskshismothertoget5Game-Boycartridgesfromhiscollectionof10cartridgesand5sports
games.Howmanywaysaretherethathismothercanget3cartridgesand2sportsgames?
Solution:Thenumberofwaysofselecting3cartridgesfrom10is:
??????
??????
??????=
??????!
??????!??????−??????!
10
�
3=
10!
3!10−3!
=
7!×8×9×10
1×2×3×7!
=
(8)(9)(10)
(1)(2)(3)
=120
31

32. Example 2 (cont.):
Solution:Thenumberofwaysofselecting2cartridgesfrom5is:
??????
??????
??????=
??????!
??????!??????−??????!
5
�
2=
5!
2!5−2!
=
3!×4×5
1×2×3!
=10
32

33. Example 3:
•Fromastudentclubconsistingof50people.Howmanydifferentchoicesofofficersarepossibleif:
a)3membersarechosenrandomly.
b)presidentandatreasureraretobechosen.
Solution:a)Given�=50,�=3,since3membersaretobechosenatrandom,itiscombination
problem.Thetotalno.ofsamplepointsare:
50
�
3=
50!
3!50−3!
=
47!×48×49×50
1×2×3×47!
=
(48)(49)(50)
(1)(2)(3)
=19,600
33

34. Example 3 (cont,):
b)presidentandatreasureraretobechosen.
Solution:Given�=50,�=2,sincepresidentandtreasureraredistinguishable,itispermutation
problem.Thetotalno.ofsamplepointsare:
50
??????
2=
50!
50−2!
=
48!×49×50
48!
=4950=2450
34

35. Practice Questions:
1.Howmanywaysaretheretoselect3candidatesfrom8equallyqualifiedrecentgraduatesfor
openingsinanaccountingfirm?
2.Inhowmanydifferentwayscanatrue-falsetestconsistingof9questionsbeanswered?
3.Adrugforthereliefofasthmacanbepurchasedfrom5differentmanufacturersinliquid,tablet,or
capsuleform,allofwhichcomeinregularandextrastrength.Howmanydifferentwayscanadoctor
prescribethedrugforapatientsufferingfromasthma?
35

36. 36
ANY QUESTION