Class 12 Maths Chapter 13 Probability – Exercise 13.5 NCERT Solutions

Introduction

Exercise 13.5 focuses on Bayes’ theorem. Students learn how to calculate conditional probabilities when multiple hypotheses are possible, apply the theorem to real‑life problems, and understand its importance in decision‑making. This exercise is crucial for mastering advanced probability concepts.

Key Concepts

1. Bayes’ Theorem: If $A_1,A_2,\dots ,A_n$ are mutually exclusive and exhaustive events, and $B$ is any event, then

$$P(A_i\mathrm{\mid }B)=\frac{P(A_i)\cdot P(B\mathrm{\mid }{A}_i)}{\sum _{j=1}^{n}P(A_j)\cdot P(B\mathrm{\mid }{A}_j)}$$

2. Prior Probability: Probability before new information is considered.

3. Posterior Probability: Probability after considering new information.

Students Frequently Make Mistakes

• Forgetting denominator in Bayes’ theorem.

• Confusing prior and posterior probabilities.

• Errors in identifying exhaustive events.

• Misinterpreting conditional probabilities.

• Skipping normalization step in denominator.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. A factory has 3 machines A, B, C producing 25%, 35%, 40% of items. Defective rates: 1%, 1.5%, 2%. Find probability defective item from machine A.

$$P(A\mathrm{\mid }D)=\frac{0.25\cdot 0.01}{0.25\cdot 0.01+0.35\cdot 0.015+0.40\cdot 0.02}=\frac{0.0025}{0.0025+0.00525+0.008}=0.156$$

Q2. A factory has 2 machines A, B producing 60%, 40% items. Defective rates: 2%, 3%. Find probability defective item from machine A.

$$P(A\mathrm{\mid }D)=\frac{0.6\cdot 0.02}{0.6\cdot 0.02+0.4\cdot 0.03}=\frac{0.012}{0.012+0.012}=0.5$$

Q3. A factory has 2 machines A, B producing 70%, 30% items. Defective rates: 4%, 5%. Find probability defective item from machine A.

$$P(A\mathrm{\mid }D)=\frac{0.7\cdot 0.04}{0.7\cdot 0.04+0.3\cdot 0.05}=\frac{0.028}{0.028+0.015}=0.651$$

Q4. A factory has 3 machines A, B, C producing 20%, 30%, 50% items. Defective rates: 2%, 3%, 4%. Find probability defective item from machine B.

$$P(B\mathrm{\mid }D)=\frac{0.3\cdot 0.03}{0.2\cdot 0.02+0.3\cdot 0.03+0.5\cdot 0.04}=\frac{0.009}{0.004+0.009+0.02}=0.273$$

Q5. A factory has 2 machines A, B producing 50%, 50% items. Defective rates: 1%, 2%. Find probability defective item from machine B.

$$P(B\mathrm{\mid }D)=\frac{0.5\cdot 0.02}{0.5\cdot 0.01+0.5\cdot 0.02}=\frac{0.01}{0.015}=0.667$$

Q6. A company has 3 suppliers A, B, C supplying 30%, 40%, 30% items. Defective rates: 3%, 4%, 5%. Find probability defective item from supplier C.

$$P(C\mathrm{\mid }D)=\frac{0.3\cdot 0.05}{0.3\cdot 0.03+0.4\cdot 0.04+0.3\cdot 0.05}=\frac{0.015}{0.009+0.016+0.015}=0.326$$

Q7. A company has 2 suppliers A, B supplying 40%, 60% items. Defective rates: 2%, 3%. Find probability defective item from supplier A.

$$P(A\mathrm{\mid }D)=\frac{0.4\cdot 0.02}{0.4\cdot 0.02+0.6\cdot 0.03}=\frac{0.008}{0.008+0.018}=0.308$$

Q8. A company has 2 suppliers A, B supplying 30%, 70% items. Defective rates: 1%, 2%. Find probability defective item from supplier B.

$$P(B\mathrm{\mid }D)=\frac{0.7\cdot 0.02}{0.3\cdot 0.01+0.7\cdot 0.02}=\frac{0.014}{0.003+0.014}=0.824$$

Q9. A company has 3 suppliers A, B, C supplying 25%, 25%, 50% items. Defective rates: 2%, 3%, 4%. Find probability defective item from supplier A.

$$P(A\mathrm{\mid }D)=\frac{0.25\cdot 0.02}{0.25\cdot 0.02+0.25\cdot 0.03+0.5\cdot 0.04}=\frac{0.005}{0.005+0.0075+0.02}=0.167$$

Q10. A company has 2 suppliers A, B supplying 55%, 45% items. Defective rates: 2%, 4%. Find probability defective item from supplier B.

$$P(B\mathrm{\mid }D)=\frac{0.45\cdot 0.04}{0.55\cdot 0.02+0.45\cdot 0.04}=\frac{0.018}{0.011+0.018}=0.621$$

FAQs (10)

FAQ1. What is Bayes’ theorem? Formula to calculate conditional probability with multiple hypotheses.

FAQ2. What is prior probability? Probability before new information.

FAQ3. What is posterior probability? Probability after considering new information.

FAQ4. What is denominator in Bayes’ theorem? Sum of all weighted probabilities.

FAQ5. What is probability defective item from machine A in Q1? 0.156.

FAQ6. What is probability defective item from machine A in Q2? 0.5.

FAQ7. What is probability defective item from machine A in Q3? 0.651.

FAQ8. What is probability defective item from machine B in Q4? 0.273.

FAQ9. What is probability defective item from supplier B in Q8? 0.824.

FAQ10. Why is Exercise 13.5 important? It builds mastery of Bayes’ theorem and conditional probability.

Conclusion

Exercise 13.5 has 10 solved questions and 10 FAQs that strengthen your understanding of Bayes’ theorem and conditional probability. This completes the Probability chapter in Class 12 Maths.

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