Probability
Board: CBSE |Class:Class11
ComprehensivestudynotesforProbabilitybyAjayYadav(MathKingofKatargam).
SamplespaceS=setofallpossibleoutcomes.EventE=subsetofS.P(E)=n(E)/n(S).0≤P(E)≤1.
Impossible:P(E)=0.Sure:P(E)=1.Complementary:P(notE)=1-P(E).Mutuallyexclusive:A∩B=φ.Exhaustive:A∪B=S.
P(S)=1.0≤P(E)≤1.IfAandBaremutuallyexclusive:P(A∪B)=P(A)+P(B).
P(A∪B)=P(A)+P(B)-P(A∩B).Forthreeevents:P(A∪B∪C)=P(A)+P(B)+P(C)-P(A∩B)-P(B∩C)-P(C∩A)+P(A∩B∩C).
P(A|B) = P(A∩B)/P(B), P(B)≠0. Multiplication theorem: P(A∩B) = P(A)P(B|A) = P(B)P(A|B).
Independent Events
A and B are independent if P(A∩B) = P(A)P(B). Also: P(A|B) = P(A) and P(B|A) = P(B).
Important Formulas
| P(A) | P(A) = n(A)/n(S) |
| Complement | P(A') = 1 - P(A) |
| Addition | P(A∪B) = P(A)+P(B)-P(A∩B) |
| Conditional | P(A|B)=P(A∩B)/P(B) |
| Independent | P(A∩B)=P(A)P(B) |
Example1:Adieisrolled.FindP(evennumber).
Solution:E={2,4,6}.P(E)=3/6=1/2.
Example2:Twodicearerolled.FindP(sum=7).
Solution:Favorable:(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)=6.Total=36.P=6/36=1/6.
Example3:Acardisdrawnfrom52cards.FindP(redorking).
Solution:P(red)=26/52,P(king)=4/52,P(redking)=2/52.P=26/52+4/52-2/52=28/52=7/13.
- Acoinistossed3times.FindP(exactly2heads).
- Twodicearerolled.FindP(productis12).
- P(A)=0.4,P(B)=0.5,P(A∩B)=0.2.FindP(A|B) and P(A∪B).
- A bag has 4 red, 5 blue balls. Two balls drawn without replacement. Find P(both red).
- Events A and B are independent with P(A)=0.3, P(B)=0.4. Find P(A∪B).
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