Class 12 Maths Chapter 13 Probability – Exercise 13.8 NCERT Solutions

Introduction

Exercise 13.8 focuses on the Binomial Distribution. Students learn how to model random experiments with two outcomes (success/failure), calculate probabilities using binomial expansion, and derive mean and variance of binomial distributions. This exercise is essential for probability applications in statistics and data science.

Key Concepts

1. Binomial Distribution: If probability of success = $p$, failure = $q=1-p$, number of trials = $n$, then

$$P(X=r)=\left(\genfrac{}{}{0px}{}{n}{r}\right)p^r{q}^{n-r},r=0,1,2,\dots ,n$$

2. Mean:

$$E(X)=np$$

3. Variance:

$$Var(X)=npq$$

4. Standard Deviation:

$$\sigma =\sqrt{npq}$$

Students Frequently Make Mistakes

• Forgetting to use binomial coefficient $\left(\genfrac{}{}{0px}{}{n}{r}\right)$.

• Confusing $p$ and $q$.

• Errors in calculating mean and variance.

• Skipping normalization check $\sum P(X=r)=1$.

• Misinterpreting number of trials $n$.

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

Q1. A coin is tossed 3 times. Find probability of getting exactly 2 heads.

$$P(X=2)=\left(\genfrac{}{}{0px}{}{3}{2}\right){\left(\frac{1}{2}\right)}^2{\left(\frac{1}{2}\right)}^1=3\cdot \frac{1}{8}=\frac{3}{8}$$

Q2. A coin is tossed 4 times. Find probability of getting exactly 3 heads.

$$P(X=3)=\left(\genfrac{}{}{0px}{}{4}{3}\right){\left(\frac{1}{2}\right)}^3{\left(\frac{1}{2}\right)}^1=4\cdot \frac{1}{16}=\frac{1}{4}$$

Q3. A die is thrown 5 times. Find probability of getting exactly 2 sixes.

$$P(X=2)=\left(\genfrac{}{}{0px}{}{5}{2}\right){\left(\frac{1}{6}\right)}^2{\left(\frac{5}{6}\right)}^3=10\cdot \frac{1}{36}\cdot \frac{125}{216}=\frac{1250}{7776}$$

Q4. A coin is tossed 6 times. Find probability of getting exactly 4 heads.

$$P(X=4)=\left(\genfrac{}{}{0px}{}{6}{4}\right){\left(\frac{1}{2}\right)}^4{\left(\frac{1}{2}\right)}^2=15\cdot \frac{1}{64}=\frac{15}{64}$$

Q5. A coin is tossed 5 times. Find probability of getting at least 3 heads.

$$P(X\ge 3)=P(X=3)+P(X=4)+P(X=5)$$

$$=\left(\genfrac{}{}{0px}{}{5}{3}\right)\frac{1}{8}\frac{1}{4}+\left(\genfrac{}{}{0px}{}{5}{4}\right)\frac{1}{16}+\left(\genfrac{}{}{0px}{}{5}{5}\right)\frac{1}{32}=\frac{16}{32}=\frac{1}{2}$$

Q6. A die is thrown 4 times. Find probability of getting no six.

$$P(X=0)=\left(\genfrac{}{}{0px}{}{4}{0}\right){\left(\frac{1}{6}\right)}^0{\left(\frac{5}{6}\right)}^4={\left(\frac{5}{6}\right)}^4=\frac{625}{1296}$$

Q7. A coin is tossed 10 times. Find mean number of heads.

$$E(X)=np=10\cdot \frac{1}{2}=5$$

Q8. A coin is tossed 10 times. Find variance of number of heads.

$$Var(X)=npq=10\cdot \frac{1}{2}\cdot \frac{1}{2}=2.5$$

Q9. A die is thrown 12 times. Find mean number of sixes.

$$E(X)=np=12\cdot \frac{1}{6}=2$$

Q10. A die is thrown 12 times. Find variance of number of sixes.

$$Var(X)=npq=12\cdot \frac{1}{6}\cdot \frac{5}{6}=\frac{60}{36}=\frac{5}{3}$$

FAQs (10)

FAQ1. What is binomial distribution formula? $P(X=r)=\left(\genfrac{}{}{0px}{}{n}{r}\right)p^r{q}^{n-r}$.

FAQ2. What is mean of binomial distribution? $E(X)=np$.

FAQ3. What is variance of binomial distribution? $Var(X)=npq$.

FAQ4. What is standard deviation? $\sigma =\sqrt{npq}$.

FAQ5. What is probability of 2 heads in 3 tosses? $\frac{3}{8}$.

FAQ6. What is probability of 3 heads in 4 tosses? $\frac{1}{4}$.

FAQ7. What is mean number of heads in 10 tosses?

FAQ8. What is variance of heads in 10 tosses? 2.5.

FAQ9. What is mean number of sixes in 12 throws?

FAQ10. Why is Exercise 13.8 important? It builds mastery of binomial distribution and its applications.

Conclusion

Exercise 13.8 has 10 solved questions and 10 FAQs that strengthen your understanding of binomial distribution. This completes the Probability chapter in Class 12 Maths.

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