Maths Class 12 Probability Project Presentation

Class 12 Maths Presentation on Probability. Mathematics Project Class 12th on Chapter 13 Probability. Project in English. Chapter 13 Up to 2020-21 Revised syllabus up to Baye's Theorem.

1. Probability
ArwachinBhartiBhawanSr.Sec.School
Aaditya Pandey Class:XII -D

2. Introduction
Probabilityisthebranchofmathematicsconcerning
numericaldescriptionsofhowlikelyaneventistooccur,
orhowlikelyitisthatapropositionistrue.
Probability=
������??????�??????��??????���������??????���
????????????���������??????�����??????���
Example:Lettherebeabasketwith3balls:Red,Green,
Blue.Ifyouwanttopickaredball,wecancalculatethe
probabilityofpickingaredball.
Totalnumberofpossibilities=3
Totalnumberoffavorablepossibility=1
Therefore,Probabilityofgettingaredball=1/3

3. Introduction
Terms Related to Probability
Sample Space: Set of all possible outcomes of a random experiment is called Sample space.
It is denoted by S.
Sample Point:Each element of the sample space is called the sample point.
Event: It is the set of favorable outcome.
Mutually Exclusive Events:Two events are said to be mutually exclusive if
there is no common element between them.
Exhaustive Events: The given events are exhaustive if when I take the elements in those
events forms the given sample space.

4. Conditional Probability
WhenIthrowadice,whatisthe
probabilitythattheoutcomeis3given
thattheoutcomeisodd?
Theansweris1/3astheoddnumbersare
1,3and5.Soweneedtheprobabilityof
gettingthenumber3outofthenumbers
1,3,5.Thechanceisoneoutofthreeso
Theprobabilityis1/3.
Note:ButP(3)=1/6Ifthereisno
condition.Astheprobabilityofgettinga
singlenumbernamed3,outoftotal6
numbersnamely1,2,3,4,5,6
is1/6

5. Conditional Probability
Definition
ProbabilityofeventEiscalledthe
conditionalprobabilityofFgiventhatEhas
alreadyoccurred,andit'sdefinedbyP(A|B).
Formulaforconditionalprobability,P(A|B)=
�(�∩�)
�(�)

6. Conditional Probability
Example:InasurveyinaclassitwasfoundthattheprobabilityofastudentwatchingYoutube
videosis0.8.andtheprobabilitythatastudentisbothtopperandalsowatchesYoutubevideos
is0.792.whatistheprobabilitythatastudentisatopperifhewatchesYoutubevideos?
Solution:LetEdenotestheeventthatapersonwatchesYoutubevideos
LetFdenotestheeventthatapersonistopper.
Then,P(E)=probabilitythatapersonwatchesYoutubevideos=0.8
P(E∩F)=probabilitythatoneisbothtopperandalsowatchesYoutubevideos=0.792
andP(F|E)=theprobabilitythatapersonisatopperifhewatchesYoutube
videos
AccordingtoFormulaforconditionalprobability,P(F|E)=P(F∩E)/P(E)
=0.792/0.8=0.99
∴TheprobabilitythatapersonisatopperifhewatchesYoutubevideosis0.99

7. Conditional Probability
Properties of Conditional Probability
P(S|F)=1
P(F|F)=1
P((A∪B)|F)=P(A|F)+P(B|F)–P((A∩B)|F)
P(E'|F)=1-P(E|F)
Example:EvaluateP(A∪B),if2P(A)=P(B)=5/13andP(A|B)=2/5
Solution:2P(A)=P(B)=5/13
=>P(B)=5/13andP(A)=1/2xP(B)=1/2×5/13=5/26
Now,theformulaofconditionalprobabilityisP(A|B)=P(A∩B)/P(B)
∴2/5=P(A∩B)/P(B)
=>P(A∩B)=2/5XP(B)=2/5×5/13=2/13
Here,theformulaisP(A∪B)=P(A)+P(B)−P(A∩B)
=>P(A∪B)=5/26+5/13-2/13=(5+10−4)/26=11/26

8. Conditional Probability
Example:IfAandBareeventssuchthatP(A|B)=P(B|A),then
(A)A⊂BbutA≠B
(B)A=B
(C)A∩B=Φ
(D)P(A)=P(B)
Solution:
ButweknowP(A|B)=P(A∩B)/P(B)
andP(B|A)=P(A∩B)/P(A)
sinceP(A|B)=P(B|A)
=>P(A∩B)/P(B)=P(A∩B)/P(A)
=>P(A)=P(B)
Thusthecorrectoptionis(D).

9. Probability
Multiplication Theorem on Probability
LetEandFbetwoeventsassociatedwithasample
spaceofanexperiment.
Then
P(E∩F)=P(E)P(F|E),P(E)≠0
=P(F)P(E|F),P(F)≠0
IfE,FandGarethreeeventsassociatedwithasample
space,then
P(E∩F∩G)=P(E)P(F|E)P(G|E∩F)

10. Independent Events
LetEandFbetwoeventsassociatedwithasamplespaceS.Iftheprobabilityofoccurrenceof
oneofthemisnotaffectedbytheoccurrenceoftheother,thenwesaythatthetwoeventsare
independent.Thus,twoeventsEandFwillbeindependent,if
P(F|E)=P(F),providedP(E)≠0
P(E|F)=P(E),providedP(F)≠0
Usingthemultiplicationtheoremonprobability,wehave
P(E∩F)=P(E)P(F)
ThreeeventsA,BandCaresaidtobemutuallyindependentifallthefollowingconditionshold:
P(A∩B)=P(A)P(B)
P(A∩C)=P(A)P(C)
P(B∩C)=P(B)P(C)
P(A∩B∩C)=P(A)P(B)P(C)

11. Baye’s Theorem
IfE1,E2,...,Enaremutuallyexclusive
andexhaustiveeventsassociated
withasamplespace,andAisany
eventofnonzeroprobability,then
??????(??????
??????|??????)=
??????(??????
??????)??????(??????|??????
??????)
σ??????(??????
??????)??????(??????|??????
??????)

12. Baye’s Theorem
Example:Abagcontains4redand4blackballs,anotherbagcontains2redand6blackballs.Oneofthe
twobagsisselectedatrandomandaballisdrawnfromthebagwhichisfoundtobered.Findthe
probabilitythattheballisdrawnfromthefirstbag.
Solution:LetE1=theeventofselectingfirstbag.
E2=theeventofselectingsecondbag.
A=theeventofgettingredball.
Sincethereisequalchanceofselectingfirstbagorselectingsecondbag,
P(E1)=P(E2)=1/2
nowP(A|E1)=P(Drawingaredballfromfirstbag)=4/8
andP(A|E2)=P(Drawingaredballfromsecondbag)=2/8=1/4
ProbabilitythatballisdrawnfromthefirstbaggiventhattheballdrawnisredisP(E1|A)
??????(??????1|??????)=
�(�)�(�|�)
�(�)�(�|�)+�(�’)�(�|�’)
=
1
2
×
1
2
(
1
2
×
1
2
+
1
2
×
1
4
)
=
2
3

13. Probability
Thank You
ArwachinBhartiBhawanSr.Sec.School
Aaditya Pandey Class:XII -D