Probability
Board: CBSE | Class: Class 12
Comprehensive study notes for Probability by Ajay Yadav.
Key Concepts
Conditional Probability
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).
Bayes' Theorem
P(EjA)=P(Ej)P(AEj)/ΣP(Ei)P(A|Ei).Usedforreverseprobability:findcausegiveneffect.
RandomVariables
ArandomvariableXassignsanumericalvaluetoeachoutcome.Probabilitydistribution:P(X=xj)=pj.Σpj=1.
MeanandVariance
Mean:μ=Σxjpj.Variance:σ²=Σ(xj-μ)²pj=Σxj²pj-μ².
BinomialDistribution
Conditions:nfixedtrials,independent,twooutcomes(success/failure),constantp.P(X=r)=ⁿCₔpkqⁿ⁻ₔ.Mean=np.Variance=npq.
ImportantFormulas
| Conditional | P(A|B) = P(A∩B)/P(B) |
| Bayes | P(EjA)=P(Ej)P(AEj)/ΣP(Ei)P(A|Ei) |
| Binomial | P(X=r)=ⁿCₔpkqⁿ⁻ₔ |
| Binomialmean | μ=np |
| Binomialvariance | σ²=npq |
SolvedExamples
Example1:IfP(A)=0.6,P(B)=0.4,P(A∩B)=0.2,findP(A|B).
Solution: P(A|B)=0.2/0.4=0.5.
Example2:Abaghas3red,2blueballs.Twodrawnwithoutreplacement.FindP(bothred).
Solution:P(R₁)P(R₂|R1) = (3/5)(2/4) = 6/20 = 3/10.
Example 3: Find mean and variance of binomial with n=5, p=0.3.
Solution: Mean=5(0.3)=1.5. Variance=5(0.3)(0.7)=1.05.
Practice Questions
- P(A)=0.5, P(B)=0.3, P(A∪B)=0.7. Find P(A|B) and P(B|A).
- A coin tossed 6 times. Find P(exactly 4 heads).
- A die rolled 4 times. Find P(getting 6 at least once).
- In a factory, Machine A produces 60%, Machine B 40%. 2% of A and 1% of B are defective. Random item is defective. Find probability it came from A.
- Find probability distribution of number of heads when 3 coins are tossed.
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Video Lessons
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