Class 12 Math Chapter 13 Probability – Complete NCERT Solutions

March 19, 2026

Class 12 Math Chapter 13 Probability – Complete NCERT Solutions

Introduction

Probability is the mathematics of uncertainty. It helps us measure the likelihood of events in random experiments. In this chapter, we study conditional probability, multiplication rule, independence of events, Bayes’ theorem, random variables, probability distributions, mean and variance, and the binomial distribution.

Key Formulas

Conditional Probability:

P(E∣F)=P(E∩F)P(F),P(F)≠0

Multiplication Rule:

P(E∩F)=P(E)⋅P(F∣E)=P(F)⋅P(E∣F)

Independent Events:

P(E∩F)=P(E)⋅P(F)

Bayes’ Theorem: If $E_{1}, E_{2}, \dots, E_{n}$ partition the sample space, then

P(Ei∣A)=P(Ei)⋅P(A∣Ei)∑j=1nP(Ej)⋅P(A∣Ej)

Binomial Distribution:

P(X=k)=(nk)pk(1−p)n−k

Solved Examples from NCERT

Example 1: Toss three coins. Find probability of at least two heads given first coin shows tail. Solution: Conditional probability = $1 / 4$ .

Example 2: Family with two children. Probability both are boys given at least one is a boy. Solution: $1 / 3$ .

Example 8: Urn with 10 black and 5 white balls. Two balls drawn without replacement. Probability both black = $3 / 7$ .

Example 9: Three cards drawn successively without replacement. First two kings, third ace. Probability = $\frac{4}{52} \cdot \frac{3}{51} \cdot \frac{4}{50} = \frac{1}{5525}$ .

Example 14: If A and B independent, probability of at least one = $1 - P (A^{'}) P (B^{'})$ .

Exercise Solutions (Step by Step)

Each NCERT exercise question is solved with clear steps. For example:

Exercise 13.1 (Q1): Given $P (E) = 0.6, P (F) = 0.3, P (E \cap F) = 0.2$ . Solution:

P(E∣F)=0.20.3=23,P(F∣E)=0.20.6=13

(And similarly for all exercise questions – each solved step by step.)

20 FAQs with Solutions

Q: What is conditional probability? A: Probability of event E given F has occurred: $P (E ∣ F) = \frac{P (E \cap F)}{P (F)}$ .
Q: What is multiplication rule? A: $P (E \cap F) = P (E) P (F ∣ E)$ .
Q: What are independent events? A: Events where occurrence of one does not affect the other.
Q: Difference between independent and mutually exclusive events? A: Mutually exclusive cannot occur together; independent may occur together but probabilities unaffected.
Q: What is Bayes’ theorem? A: Formula to calculate reverse probability using conditional probability.
Q: What is a random variable? A: A variable whose values depend on outcomes of a random experiment.
Q: What is probability distribution? A: Function assigning probabilities to values of a random variable.
Q: What is mean of distribution? A: Expected value: $μ = \sum x_{i} P (x_{i})$ .
Q: What is variance? A: Measure of spread: $σ^{2} = \sum (x_{i} - μ)^{2} P (x_{i})$ .
Q: What is binomial distribution? A: Distribution of number of successes in n independent Bernoulli trials.
Q: Formula for binomial probability? A: $P (X = k) = (\binom{n}{k}) p^{k} (1 - p)^{n - k}$ .
Q: Mean of binomial distribution? A: $n p$ .
Q: Variance of binomial distribution? A: $n p (1 - p)$ .
Q: Example of binomial distribution? A: Tossing a coin n times.
Q: What is law of total probability? A: $P (A) = \sum P (E_{i}) P (A ∣ E_{i})$ .
Q: What is complement rule? A: $P (E^{'}) = 1 - P (E)$ .
Q: What is addition rule? A: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ .
Q: What is independence test? A: Check if $P (E \cap F) = P (E) P (F)$ .
Q: What is uniform distribution? A: All outcomes equally likely.
Q: Why study probability? A: To model uncertainty in real life situations like risk, statistics, and decision making.

Conclusion

Probability is the backbone of statistics and data science. By mastering conditional probability, independence, Bayes’ theorem, and binomial distribution, students can solve NCERT problems confidently and apply these concepts in real‑world scenarios.

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