Class 12 Maths Chapter 13 Probability – Exercise 13.9 NCERT Solutions

Introduction

Exercise 13.9 focuses on the Poisson Distribution. Students learn how to model rare events, calculate probabilities using the Poisson formula, and derive mean and variance. This exercise is essential for probability applications in statistics, especially when dealing with large trials and small success probabilities.

Key Concepts

1. Poisson Distribution Formula: If average rate of success = $\lambda$, then

$$P(X=r)=\frac{e^{-\lambda }{\lambda }^r}{r!},r=0,1,2,\dots$$

2. Mean:

$$E(X)=\lambda$$

3. Variance:

$$Var(X)=\lambda$$

Students Frequently Make Mistakes

• Forgetting to use factorial in denominator.

• Confusing Poisson with Binomial distribution.

• Errors in calculating $e^{-\lambda }$.

• Misinterpreting mean as probability.

• Skipping variance calculation.

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

Q1. If $\lambda =2$, find $P(X=0)$.

$$P(0)=\frac{e^{-2}\cdot 2^0}{0!}=e^{-2}=0.1353$$

Q2. If $\lambda =2$, find $P(X=1)$.

$$P(1)=\frac{e^{-2}\cdot 2^1}{1!}=2e^{-2}=0.2707$$

Q3. If $\lambda =2$, find $P(X=2)$.

$$P(2)=\frac{e^{-2}\cdot 2^2}{2!}=2e^{-2}=0.2707$$

Q4. If $\lambda =2$, find $P(X=3)$.

$$P(3)=\frac{e^{-2}\cdot 2^3}{3!}=\frac{8}{6}{e}^{-2}=1.333e^{-2}=0.1804$$

Q5. If $\lambda =2$, find $P(X=4)$.

$$P(4)=\frac{e^{-2}\cdot 2^4}{4!}=\frac{16}{24}{e}^{-2}=0.667e^{-2}=0.0902$$

Q6. If $\lambda =3$, find $P(X=0)$.

$$P(0)=e^{-3}=0.0498$$

Q7. If $\lambda =3$, find $P(X=1)$.

$$P(1)=\frac{e^{-3}\cdot 3}{1!}=3e^{-3}=0.1494$$

Q8. If $\lambda =3$, find $P(X=2)$.

$$P(2)=\frac{e^{-3}\cdot 9}{2}=4.5e^{-3}=0.2240$$

Q9. If $\lambda =3$, find $P(X=3)$.

$$P(3)=\frac{e^{-3}\cdot 27}{6}=4.5e^{-3}=0.2240$$

Q10. If $\lambda =3$, find $P(X=4)$.

$$P(4)=\frac{e^{-3}\cdot 81}{24}=3.375e^{-3}=0.1680$$

FAQs (10)

FAQ1. What is Poisson distribution formula? $P(X=r)=\frac{e^{-\lambda }{\lambda }^r}{r!}$.

FAQ2. What is mean of Poisson distribution? $\lambda$.

FAQ3. What is variance of Poisson distribution? $\lambda$.

FAQ4. What is probability of 0 events when $\lambda =2$? 0.1353.

FAQ5. What is probability of 1 event when $\lambda =2$? 0.2707.

FAQ6. What is probability of 2 events when $\lambda =2$? 0.2707.

FAQ7. What is probability of 3 events when $\lambda =2$? 0.1804.

FAQ8. What is probability of 0 events when $\lambda =3$? 0.0498.

FAQ9. What is probability of 2 events when $\lambda =3$? 0.2240.

FAQ10. Why is Exercise 13.9 important? It builds mastery of Poisson distribution and rare event modeling.

Conclusion

Exercise 13.9 has 10 solved questions and 10 FAQs that strengthen your understanding of Poisson distribution. This completes the Probability chapter in Class 12 Maths.

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