Class 12 Maths Chapter 13 Probability – Exercise 13.4 NCERT Solutions

Introduction

Exercise 13.4 focuses on the multiplication theorem of probability and independence of events. Students learn how to calculate joint probabilities, apply conditional probability, and verify independence of events. This exercise is essential for mastering probability theory and preparing for Bayes’ theorem.

Key Concepts

1. Multiplication Theorem: For any two events $A$ and $B$,

$$P(A\cap B)=P(A\mathrm{\mid }B)\cdot P(B)=P(B\mathrm{\mid }A)\cdot P(A)$$

2. Independent Events: If $P(A\cap B)=P(A)\cdot P(B)$, then $A$ and $B$ are independent.

3. Conditional Probability Reminder:

$$P(A\mathrm{\mid }B)=\frac{P(A\cap B)}{P(B)},P(B)\mathrm{\ne }0$$

Students Frequently Make Mistakes

• Forgetting to check independence condition properly.

• Confusing $P(A\mathrm{\mid }B)$ with $P(B\mathrm{\mid }A)$.

• Errors in multiplication theorem application.

• Skipping denominator in conditional probability.

• Misinterpreting independence as mutually exclusive.

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

Q1. If $P(A)=\frac{1}{2},P(B)=\frac{1}{3},P(A\cap B)=\frac{1}{6}$, are $A$ and $B$ independent?

$$P(A)\cdot P(B)=\frac{1}{2}\cdot \frac{1}{3}=\frac{1}{6}=P(A\cap B)$$

Yes, independent.

Q2. If $P(A)=0.4,P(B)=0.5,P(A\cap B)=0.2$, are $A$ and $B$ independent?

$$P(A)\cdot P(B)=0.4\cdot 0.5=0.2=P(A\cap B)$$

Yes, independent.

Q3. If $P(A)=0.5,P(B)=0.6,P(A\cap B)=0.25$, are $A$ and $B$ independent?

$$P(A)\cdot P(B)=0.5\cdot 0.6=0.3\mathrm{\ne }0.25$$

No, not independent.

Q4. If $P(A)=0.3,P(B)=0.4,P(A\cap B)=0.12$, are $A$ and $B$ independent?

$$P(A)\cdot P(B)=0.3\cdot 0.4=0.12=P(A\cap B)$$

Yes, independent.

Q5. If $P(A)=0.7,P(B)=0.5,P(A\cap B)=0.35$, are $A$ and $B$ independent?

$$P(A)\cdot P(B)=0.7\cdot 0.5=0.35=P(A\cap B)$$

Yes, independent.

Q6. If $P(A)=0.6,P(B)=0.3,P(A\cap B)=0.18$, are $A$ and $B$ independent?

$$P(A)\cdot P(B)=0.6\cdot 0.3=0.18=P(A\cap B)$$

Yes, independent.

Q7. If $P(A)=0.2,P(B)=0.5,P(A\cap B)=0.15$, are $A$ and $B$ independent?

$$P(A)\cdot P(B)=0.2\cdot 0.5=0.1\mathrm{\ne }0.15$$

No, not independent.

Q8. If $P(A)=0.4,P(B)=0.6,P(A\cap B)=0.24$, are $A$ and $B$ independent?

$$P(A)\cdot P(B)=0.4\cdot 0.6=0.24=P(A\cap B)$$

Yes, independent.

Q9. If $P(A)=0.5,P(B)=0.5,P(A\cap B)=0.3$, are $A$ and $B$ independent?

$$P(A)\cdot P(B)=0.25\mathrm{\ne }0.3$$

No, not independent.

Q10. If $P(A)=0.25,P(B)=0.4,P(A\cap B)=0.1$, are $A$ and $B$ independent?

$$P(A)\cdot P(B)=0.25\cdot 0.4=0.1=P(A\cap B)$$

Yes, independent.

FAQs (10)

FAQ1. What is multiplication theorem of probability? $P(A\cap B)=P(A\mathrm{\mid }B)\cdot P(B)$.

FAQ2. What is conditional probability formula? $P(A\mathrm{\mid }B)=\frac{P(A\cap B)}{P(B)}$.

FAQ3. What are independent events? Events where $P(A\cap B)=P(A)\cdot P(B)$.

FAQ4. Are independent events same as mutually exclusive? No, mutually exclusive means $P(A\cap B)=0$.

FAQ5. What is probability of intersection of independent events? Product of their probabilities.

FAQ6. What is solution of Q1? Yes, independent.

FAQ7. What is solution of Q3? No, not independent.

FAQ8. What is solution of Q5? Yes, independent.

FAQ9. What is solution of Q7? No, not independent.

FAQ10. Why is Exercise 13.4 important? It builds mastery of multiplication theorem and independence of events.

Conclusion

Exercise 13.4 has 10 solved questions and 10 FAQs that strengthen your understanding of multiplication theorem and independence of events. This builds the foundation for Bayes’ theorem in Class 12 Maths.

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