Summary
- Law of total probability: P(A)=ΣP(A|Bk)P(Bk)
- Bayes' theorem: P(B|A)=P(A|B)P(B)/P(A)
- Independence: P(A∩B)=P(A)P(B)
- Binomial distribution: E=np、V=np(1-p)
- Normal distribution: Standardize and use the Z-table
Comprehensive Practice Problems
- Factory X (70%) has a defect rate of 3%, and Factory Y (30%) has a defect rate of 7%. Given that a product is defective, what is the probability that it came from Factory Y?
- P(A)=0.6、P(B)=0.5、and they are independent. Find P(A∪B̄).
- What is the probability of getting at least one 6 when a die is rolled three times?
Answers
- P(defective)=0.7×0.03+0.3×0.07=0.042。P(Y|defective)=0.021/0.042=1/2
- P(B̄)=0.5。A and B̄ are also independent. P(A∩B̄)=0.3。P(A∪B̄)=0.6+0.5−0.3=0.8
- 1−(5/6)³=91/216