CBSE · Class 12 · Mathematics · Chapter 13

Chapter 13
Probability

Complete chapter resources for CBSE Class 12 Maths — topic breakdown, key formulas, sample questions, previous year board questions, and instant AI question paper generation.

4Topics
8–10Board marks
8Sample questions
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Key Formulas — Chapter 13
  • Conditional probability: P(A|B) = P(A ∩ B) / P(B), P(B) ≠ 0
  • Multiplication rule: P(A ∩ B) = P(A) · P(B|A)
  • Independence: P(A ∩ B) = P(A) · P(B)
  • Total probability: P(E) = Σ P(Hᵢ) · P(E|Hᵢ)
  • Bayes' theorem: P(Hᵢ|E) = P(Hᵢ)·P(E|Hᵢ) / Σ P(Hⱼ)·P(E|Hⱼ)
  • Binomial distribution: P(X=r) = ⁿCᵣ · pʳ · qⁿ⁻ʳ
Chapter Overview

What this chapter covers

Chapter 13 of NCERT Class 12 Mathematics builds on the foundational probability concepts from Class 11 and introduces a more rigorous framework. The chapter opens with conditional probability — the probability of event A given that event B has already occurred — and uses it to define the multiplication rule. A key distinction drawn early is between independent events (where P(A ∩ B) = P(A) · P(B)) and mutually exclusive events, a common source of board exam errors.

The chapter then introduces the theorem of total probability and Bayes' theorem, which allow students to work backwards from an observed outcome to determine which of several possible causes is most likely. These theorems are routinely tested as 5-mark long-answer questions involving real-world scenarios such as manufacturing defects, disease testing, or quality control. Step-by-step tabular calculation is the expected method in board answers.

The final section covers random variables and their probability distributions, including the mean (expectation) and variance. The chapter concludes with the Binomial distribution — a specific distribution for repeated independent Bernoulli trials — giving its probability mass function, mean (np), and variance (npq). Board questions on random variables frequently ask for a complete distribution table followed by E(X) and Var(X) calculations.

Topics

What's inside Chapter 13

As per NCERT Class 12 Mathematics (CBSE syllabus)

Topic 1
Conditional Probability & Multiplication Rule
P(A|B) = P(A ∩ B) / P(B). Properties of conditional probability. Multiplication theorem for two and more events. Independent vs. mutually exclusive events.
Topic 2
Theorem of Total Probability
Partition of a sample space into mutually exclusive and exhaustive events. P(E) = Σ P(Hᵢ) · P(E|Hᵢ). Setting up prior probabilities for multi-cause problems.
Topic 3
Bayes' Theorem
Computing posterior probabilities from prior and likelihood. P(Hᵢ|E) formula. Systematic tabular approach for board exam solutions. Applications in quality control, medical diagnosis.
Topic 4
Random Variables, Distributions & Binomial Distribution
Discrete random variable, probability distribution table. E(X) = Σ xᵢP(xᵢ), Var(X). Bernoulli trials and the Binomial distribution: P(X=r) = ⁿCᵣ pʳ qⁿ⁻ʳ, mean = np, variance = npq.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Class 11 · Probability
Sample space, events, classical and axiomatic definitions of probability
Class 11 · Permutations & Combinations
Counting principles used to enumerate favourable and total outcomes
Chapter 13 Probability
Leads to
Class 12 · Statistics (Applied Maths)
Probability distributions, hypothesis testing, confidence intervals
JEE / Engineering Entrance
Higher-order combinatorial probability, geometric probability, Poisson distribution
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 12 Maths board papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Assertion-Reason 1 1–2 Conditional probability value, independence check, or basic distribution property
Very Short Answer 2 1 Conditional probability, P(A ∪ B) when independent, or E(X) of a simple distribution
Short Answer 3 1 Total probability theorem or Binomial distribution (mean / variance)
Long Answer — Bayes / Random Variable 5 1 Bayes' theorem word problem, or full probability distribution table with E(X) and Var(X)
Total (approximate) 8–10 4–5 Weightage varies across paper sets and years
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 12 Maths Chapter 13. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
If P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2, then P(A | B) is: (a) 0.2 (b) 0.4 (c) 0.5 (d) 0.8
Q2 Easy 2 marks Short Answer
Events A and B are such that P(A) = 1/2, P(B) = 7/12, and P(not A or not B) = 1/4. State whether A and B are independent. Justify your answer.
Q3 Medium 2 marks Short Answer
A bag contains 4 red and 6 black balls. Two balls are drawn without replacement. Find the probability that both balls drawn are red.
Q4 Medium 3 marks Short Answer
A die is thrown three times. Let X denote the number of times an odd number appears. Write down the probability distribution of X and find E(X).
Q5 Medium 3 marks Word Problem
In a factory, machines A, B, and C produce 50%, 30%, and 20% of the total output respectively. Their defective rates are 3%, 4%, and 5%. An item is chosen at random. Using the theorem of total probability, find the probability that the chosen item is defective.
Q6 Hard 5 marks Word Problem
In the factory problem above (Q5), an item is found to be defective. Using Bayes' theorem, find the probability that it was produced by: (i) Machine A (ii) Machine B (iii) Machine C
Q7 Hard 5 marks Word Problem
Two cards are drawn successively with replacement from a well-shuffled pack of 52 cards. Let X denote the number of aces drawn. (i) Write down the probability distribution of X. (ii) Calculate the mean E(X) and variance Var(X) of the distribution.
Q8 Hard 4 marks Word Problem
The probability that a student passes Mathematics is 2/3 and the probability that the same student passes Statistics is 4/9. If the probability of passing at least one subject is 4/5, find the probability that the student passes both subjects. Are the events of passing Mathematics and Statistics independent?
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Previous Year Questions

From CBSE board examinations

Actual questions from past Class 12 Maths board papers — Probability chapter.

Board 20222 marks
A and B are two events such that P(A) = 0.54, P(B) = 0.69, and P(A ∩ B) = 0.35. Find P(A' ∩ B'). (CBSE 2022, Delhi)
Board 20235 marks
Bag I contains 3 red and 4 black balls, and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II at random, and then a ball is drawn from Bag II. The ball drawn is found to be red in colour. Find the probability that the transferred ball is also red. (CBSE 2023, All India)
Board 20204 marks
The random variable X has a probability distribution P(X) of the following form, where k is some number: P(X = 0) = k, P(X = 1) = 2k, P(X = 2) = 3k, P(X = x) = 0 otherwise. (i) Find the value of k. (ii) Find P(X < 2), P(X ≤ 2), and P(X ≥ 2). (CBSE 2020)
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FAQ

Questions teachers ask

How many marks does Probability carry in the CBSE Class 12 Maths board exam? +
Probability typically carries 8–10 marks in the CBSE Class 12 Maths board paper, spread across 2–3 questions — usually one 2-mark short answer on conditional probability or independence, one 3-mark question on Bayes' theorem or total probability, and one 5-mark long answer involving a random variable or Binomial distribution. The chapter has appeared in every CBSE Class 12 board paper consistently.
What is the difference between independent events and mutually exclusive events in Probability? +
Two events A and B are independent if the occurrence of one does not affect the other: P(A ∩ B) = P(A) · P(B). Two events are mutually exclusive if they cannot occur simultaneously: P(A ∩ B) = 0. Mutually exclusive events (unless one has probability 0) are NOT independent — this is a classic board-exam trap. Students must distinguish clearly between the two definitions.
How do students apply Bayes' theorem in board exam questions? +
Bayes' theorem questions typically describe a scenario with two or more causes (hypotheses) and give conditional probabilities of an observed effect. The formula P(Hᵢ | E) = P(Hᵢ) · P(E | Hᵢ) / Σ P(Hⱼ) · P(E | Hⱼ) is applied step-by-step. Students should always set up a table of prior probabilities and conditional likelihoods first, then compute the denominator (total probability) before finding the posterior. These are usually 5-mark questions.
What is a random variable and how is its mean (expectation) calculated? +
A random variable X assigns a numerical value to each outcome of a random experiment. Its probability distribution is a table of values xᵢ and corresponding probabilities P(X = xᵢ) such that Σ P(xᵢ) = 1. The mean (expected value) is E(X) = Σ xᵢ · P(xᵢ), and the variance is Var(X) = Σ xᵢ² · P(xᵢ) − [E(X)]². Board questions at 5 marks typically ask for the full distribution table, E(X), and Var(X).
How do I generate a custom question paper for Probability using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 12 Mathematics, select Chapter 13 (Probability), set your preferred question-type mix (MCQ, short answer, word problem) and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.