Total probability, Baye’s theorem, 20 mcqs on it with explained answers
Here are 20 MCQs on Total Probability Theorem and Bayes’ Theorem with detailed explained answers:
1. Three machines produce 30%, 50%, and 20% of bulbs. Defective rates are 2%, 3%, and 5%. What is the probability a bulb is defective?
- a) 0.034
- b) 0.03
- c) 0.04
- d) 0.05
Answer: Use total probability: $$0.3 \times 0.02 + 0.5 \times 0.03 + 0.2 \times 0.05 = 0.034$$.
2. Using above data, if a bulb is defective, find probability it came from second machine.
- a) 0.44
- b) 0.5
- c) 0.38
- d) 0.6
Answer: Use Bayes: $$\frac{0.5 \times 0.03}{0.034} = 0.44$$.
3. Sample space partitioned into $$A_1, A_2, A_3$$ with probabilities 0.2, 0.5, 0.3. $$P(B|A_1) = 0.1, P(B|A_2) = 0.4, P(B|A_3) = 0.7$$. Find $$P(B)$$.
- a) 0.43
- b) 0.4
- c) 0.39
- d) 0.41
Answer: Total Prob: $$0.2 \times 0.1 + 0.5 \times 0.4 + 0.3 \times 0.7 = 0.43$$.
4. If $$P(A) = 0.7$$, $$P(B|A) = 0.4$$, find $$P(A \cap B')$$.
- a) 0.42
- b) 0.28
- c) 0.3
- d) 0.18
Answer: $$P(A \cap B') = 0.7 \times (1-0.4) = 0.42$$.
5. A doctor diagnoses disease with 99% accuracy. Disease prevalence is 0.005. Find probability patient has disease given positive test.
- a) 0.07
- b) 0.50
- c) 0.90
- d) 0.99
Answer: Use Bayes with false positives; approximate answer 0.07.
6. Event $$B$$ partitioned by $$A$$ and $$A^c$$. If $$P(B|A)=0.6$$, $$P(A)=0.4$$, $$P(B|A^c)=0.3$$, find $$P(B)$$.
- a) 0.42
- b) 0.36
- c) 0.45
- d) 0.40
Answer: Total Prob: $$0.4 \times 0.6 + 0.6 \times 0.3 = 0.42$$.
7. If events $$A_1, A_2, A_3$$ partition the space and $$P(A_1) = 0.3, P(A_2) = 0.5, P(A_3) = 0.2$$, $$P(B|A_1) = 0.1, P(B|A_2) = 0.6, P(B|A_3) = 0.8$$, find $$P(A_3|B)$$.
- a) 0.26
- b) 0.31
- c) 0.335
- d) 0.20
Answer: Bayes: $$\frac{0.2 \times 0.8}{P(B \text{ total})} \approx 0.31$$.
8. A box has 3 urns with different white ball probabilities 0.5, 0.6, 0.8. Urns chosen equally likely. Probability of drawing white ball?
- a) 0.63
- b) 0.5
- c) 0.6
- d) 0.7
Answer: $$1/3 \times (0.5+0.6+0.8) = 0.63$$.
9. Disease test has 90% sensitivity and 99% specificity. Disease prevalence is 1%. Probability patient has disease if test positive?
- a) 0.48
- b) 0.90
- c) 0.01
- d) 0.32
Answer: Use Bayes calculating PPV ≈ 0.48.
10. $$P(A) = 0.5$$, $$P(B|A) = 0.1$$, $$P(B|A^c) = 0.6$$. Find $$P(A|B)$$.
- a) 0.15
- b) 0.48
- c) 0.27
- d) 0.50
Answer: Bayes: $$\frac{0.5 \times 0.1}{0.5 \times 0.1 + 0.5 \times 0.6} = \frac{0.05}{0.35} = 0.143$$.
11. A factory has two machines producing 70% and 30% of items. Defective rates are 5% and 10%. Probability randomly selected item is defective?
- a) 0.065
- b) 0.10
- c) 0.07
- d) 0.08
Answer: Total Prob: $$0.7 \times 0.05 + 0.3 \times 0.10 = 0.065$$.
12. Using above data, find probability defective item came from first machine.
- a) 0.54
- b) 0.65
- c) 0.45
- d) 0.35
Answer: Bayes: $$\frac{0.7 \times 0.05}{0.065} = 0.54$$.
13. Events $$A$$ and $$B$$ partition sample space with $$P(A) = 0.7$$, $$P(B) = 0.3$$. Given $$P(C|A) = 0.4$$, $$P(C|B) = 0.5$$, find $$P(C)$$.
- a) 0.43
- b) 0.44
- c) 0.45
- d) 0.42
Answer: $$0.7 \times 0.4 + 0.3 \times 0.5 = 0.43$$.
14. Probability a US citizen is a smoker is 25%. If a randomly chosen person is a smoker, probability that he lives in urban area given urban population is 60% and 30% smokers urban?
- a) 0.72
- b) 0.71
- c) 0.75
- d) 0.70
Answer: Use Bayes considering smoker and urban distribution ~0.72.
15. Three urns with varying red ball proportions 0.2, 0.5, 0.3. One urn chosen with probabilities 0.3, 0.4, 0.3. Probability randomly drawn ball is red?
- a) 0.35
- b) 0.40
- c) 0.38
- d) 0.33
Answer: Total Prob: $$0.3 \times 0.2 + 0.4 \times 0.5 + 0.3 \times 0.3 = 0.38$$.
16. If probability of defective item from machines A and B is 0.2 and 0.3 and production proportions are 40% and 60%. Find probability item is defective.
- a) 0.26
- b) 0.22
- c) 0.24
- d) 0.2
Answer: $$0.4 \times 0.2 + 0.6 \times 0.3 = 0.26$$.
17. If an item is defective, find probability it was produced by machine A using above data.
- a) 0.31
- b) 0.35
- c) 0.29
- d) 0.40
Answer: Bayes: $$\frac{0.4 \times 0.2}{0.26} = 0.31$$.
18. Probability a patient tests positive is 0.04. If prevalence of disease is 0.01 and test sensitivity 0.9, find probability patient doesn’t have disease given positive test.
- a) 0.27
- b) 0.33
- c) 0.40
- d) 0.35
Answer: Use Bayes to find false positive proportion ~0.27.
19. Probability of selecting a good quality bulb from two machines producing 25% and 75% of bulbs. Good bulb rates are 97% and 95%.
- a) 0.955
- b) 0.96
- c) 0.9555
- d) 0.96
Answer: Total Prob: $$0.25 \times 0.97 + 0.75 \times 0.95 = 0.955$$.
20. Three boxes have light bulbs with 1%, 2%, and 3% defect rates. Boxes selected with probabilities $$1/4, 1/4, 1/2$$. Probability bulb drawn is defective?
- a) 0.0225
- b) 0.02
- c) 0.025
- d) 0.03
Answer: $$ \frac{1}{4} \times 0.01 + \frac{1}{4} \times 0.02 + \frac{1}{2} \times 0.03 = 0.0225$$.
