Class 12 Maths NCERT Solutions Chapter 13 Probability | Boundless Maths
Chapter 13 Class 12 Maths NCERT Solutions — Unit VI · Probability

Class 12 Maths NCERT Solutions Chapter 13: Probability

This page brings together the Class 12 Maths NCERT Solutions Chapter 13 needs from start to finish — free, step-by-step solutions for all four parts: conditional probability, independent events, Bayes' theorem, and the Miscellaneous Exercise that mixes all three. Solved the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year, right on this page.

4Exercises (incl. Misc.)
62Total Questions
2026-27CBSE Syllabus
100%Solved

Class 12 Maths NCERT Solutions Chapter 13 — Overview

These Class 12 Maths NCERT Solutions Chapter 13 pages cover Probability, the final chapter of the Class 12 Maths NCERT textbook, which builds directly on the sample-space ideas from Class 11. Here, the question shifts from "what's the probability of an event" to "how does that probability change once you know something else has already happened" — that's conditional probability, and it's the foundation everything else in this chapter rests on.

From there, the chapter branches into independent events (where knowing one thing tells you nothing new about another), and then into Bayes' theorem — the single most exam-relevant idea here, since it lets you work backward from an observed outcome to the probability of whatever "cause" produced it. The Miscellaneous Exercise pulls all of it together, along with a touch of Bernoulli-trial reasoning, in the kind of mixed, real-world word problems that show up in Section D and Section E of the board paper.

How the Chapter Builds

One Idea Leads to the Next

1

Conditional Probability

P(E|F) = P(E∩F)/P(F) — and the multiplication rule that follows from it. Exercise 13.1.

→
2

Independent Events

When P(E∩F) = P(E)·P(F), knowing F tells you nothing new about E. Exercise 13.2.

→
3

Bayes' Theorem

Total probability across a partition of hypotheses, then reversed to find P(cause | observed effect). Exercise 13.3.

→
4

Putting It Together

Mixed problems spanning all three ideas, plus Bernoulli-trial style reasoning. Miscellaneous Exercise.

Quick Reference

Important Formulas — Chapter 13

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below have the full printable version for all of Probability.

Conditional Probability & Multiplication Rule (§13.2–13.3)

Conditional probability of E given F

P(E|F) = \dfrac{P(E\cap F)}{P(F)}, \quad P(F)\neq 0

Only defined when P(F) ≠ 0 — always check this before applying the formula.

Multiplication rule (two events)

P(E\cap F) = P(E)\cdot P(F|E) = P(F)\cdot P(E|F)

Essential for "without replacement" problems, where the second draw depends on the first.

Multiplication rule (three events)

P(E\cap F\cap G) = P(E)\cdot P(F|E)\cdot P(G|E\cap F)

Extends naturally to any number of successive, dependent events.

Independent Events (§13.4)

Independence condition

P(E\cap F) = P(E)\cdot P(F)

Equivalent to P(E|F) = P(E) and P(F|E) = P(F), whenever those are defined.

Independence of complements

If E, F are independent, then so are: E', F   and   E, F'   and   E', F'

Independent ≠ mutually exclusive — two mutually exclusive events with nonzero probability can never be independent.

Total Probability & Bayes' Theorem (§13.5)

Theorem of total probability

P(A) = \sum_{i=1}^{n} P(E_i)\cdot P(A|E_i)

Requires {E₁, E₂, ..., Eₙ} to form a genuine partition of the sample space.

Bayes' theorem

P(E_i|A) = \dfrac{P(E_i)\cdot P(A|E_i)}{\sum_{j=1}^{n} P(E_j)\cdot P(A|E_j)}

The denominator is exactly the total-probability formula above — compute it first.

Decision Guide

Which Tool Should I Use?

A quick way to decide which formula a probability word-problem is actually calling for.

SituationUse thisWhy
You're given P(E), P(F) and P(E∩F) directlyConditional Probability formulaP(E|F) = P(E∩F)/P(F) — no need for a sample space at all.
Two draws happen in sequence, without replacementMultiplication ruleThe second draw's probability genuinely depends on the first — P(E)·P(F|E), not P(E)·P(F).
You're asked to check or prove independenceIndependence test P(E∩F) = P(E)·P(F)The single defining condition — if it fails, the events are dependent, full stop.
Events are explicitly stated as independentMultiply the individual probabilities directlyP(E∩F), P(E|F) and P(F|E) all collapse to simple products once independence is given.
You need to work backward from an outcome to "which cause produced it"Bayes' TheoremReverses P(effect|cause) into P(cause|effect) using every hypothesis in the partition.
An event can happen via several different branches (bags, machines, routes)Theorem of Total Probability firstSum over every branch to get P(A) — this is also the denominator Bayes' theorem needs.
Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across all four exercises.

  • Confusing P(E|F) with P(F|E). The event you're conditioning on always sits in the denominator — swapping them is one of the most common silent errors in this chapter.
  • Assuming "independent" means "mutually exclusive." They're unrelated ideas — two mutually exclusive events with nonzero probability can never actually be independent.
  • Applying the conditional probability formula when P(F) = 0. The expression is undefined in that case, not zero — check this before dividing.
  • Treating dependent draws as independent in "without replacement" problems — using P(E)·P(F) instead of the multiplication rule P(E)·P(F|E) when the second draw's odds genuinely shift after the first.
  • Mixing up the prior with the posterior in a Bayes' theorem problem — the prior P(Eᵢ) is what's true before you know the outcome; the posterior P(Eᵢ|A) is what you're actually solving for.
  • Skipping the total probability step and guessing P(A) directly instead of summing P(Eᵢ)·P(A|Eᵢ) over every branch of the partition.
  • Recomputing "without replacement" probabilities in problems that are actually with replacement, where each draw is independent and the probabilities never change.
  • Forgetting a branch of the partition before applying Bayes' theorem — missing one bag, machine, or operator silently breaks the denominator and every answer that follows from it.
Solve Chapter-Wise

Class 12 Maths NCERT Solutions Chapter 13 — Choose an Exercise

13.1

Exercise 13.1

Conditional probability — computing P(E|F) from given probabilities and directly from sample spaces · 17 questions

Solve Exercise 13.1 →
13.2

Exercise 13.2

Independent events — testing and applying P(A∩B) = P(A)·P(B) · 18 questions

Solve Exercise 13.2 →
13.3

Exercise 13.3

Total probability and Bayes' theorem — reversing conditional probabilities to find the cause behind an outcome · 14 questions

Solve Exercise 13.3 →
M

Miscellaneous Exercise

Mixed problems combining conditional probability, independence and Bayes' theorem · 13 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Probability — conditional probability, independence, Bayes' theorem — in one printable PDF.

Get Formula Cards →

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Common Questions

Frequently Asked Questions

Quick answers about Chapter 13, Probability.

How many exercises are there in Chapter 13, Probability?

There are three main exercises — 13.1, 13.2 and 13.3 — plus a Miscellaneous Exercise, totalling 62 questions across conditional probability, independent events, and Bayes' theorem.

Is this chapter important for the board exam?

Yes — Probability carries significant weightage every year, and Bayes' theorem problems in particular are a near-certain contributor to Section D or Section E, since they combine several sub-steps worth multiple marks.

What is the difference between conditional probability and Bayes' theorem?

Conditional probability, P(E|F), moves forward — given that F has happened, what's the chance E also happens? Bayes' theorem moves backward — given that an outcome has already been observed, what's the probability it was caused by a particular hypothesis from a partition of possibilities? Bayes' theorem is really conditional probability applied in reverse, using the theorem of total probability to handle the denominator.

What should I revise before starting this chapter?

Revisit the Class 11 Probability chapter first — sample spaces, events, and the basic addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) are assumed knowledge here. This chapter builds directly on top of that foundation rather than reintroducing it.

Is Probability the last chapter in the Class 12 Maths NCERT textbook?

Yes — Chapter 13, Probability, is the final chapter of the NCERT Class 12 Mathematics textbook (Part II), coming after Chapter 12, Linear Programming.

Where can I find the official NCERT textbook for this chapter?

Probability is Chapter 13 of the NCERT Class 12 Mathematics textbook (Part II), published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website.
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