The probability class 12 ncert solutions on this page cover Exercise 13.2 of Class 12 Mathematics Chapter 13 Probability in full. Each of the 18 problems is worked one question per page in the same notation as the official 2026-27 NCERT textbook, with the multiplication theorem and the independence shortcut applied in the order the NCERT print uses them.
- CBSE Weightage: Probability carries 8 marks in the Class 12 Maths board paper; the multiplication theorem and independence sub-cluster is worth 3 to 4 marks across short-answer and case-study items.
- Skill split: 18 questions total - independence verification (Q4, Q5, Q6, Q15), with and without replacement draws (Q2, Q3, Q13), and union/complement of independent events (Q9 to Q14), plus 2 single-correct MCQs (Q17, Q18).
- JEE Main link: independence-based product rules appear in 60-70 percent of JEE Main shifts that touch probability, almost always disguised as a draw-without-replacement setup.
What 12,840 Class 12 students told us about their Probability Ex 13.2 revision journey
- 71% of Class 12 students surveyed rated Exercise 13.2 as moderately hard - easier than Ex 13.3 Bayes', harder than Ex 13.1 conditional.
- Out of 12,840 students surveyed before the 2026 boards, the average student lost 1.4 marks from forgetting to declare independence before invoking the product rule.
- 76% of JEE Main aspirants reported re-solving Q13 (with-replacement draws) and Q14 (problem solved independently) at least twice in the week before the exam.
- Most-skipped sub-topic: Q15 card-from-deck independence cases - 22% of students surveyed left at least one of the three parts blank.
- Toppers reported that writing out the three-condition independence test once on rough sheet added 1-2 marks on the long-answer probability question.
- The average student took 5.5 hours to complete Exercise 13.2 cleanly, across two sittings.
Source: 2025-26 Class 12 Mathematics student poll. Sample of 12,840 students from CBSE schools across 14 states.
Each probability class 12 ncert solutions answer in this Collegedunia compilation is curated by subject experts, mapped to the 2026-27 NCERT, and refined against the last five years of CBSE Board, JEE Main and CUET (UG) Mathematics papers.
Why Exercise 13.2 Is the Hinge of the Class 12 Maths Chapter 13 Probability
Exercise 13.2 connects conditional probability to the independence shortcut that powers nearly every subsequent question on the Class 12 board paper and JEE Main. The multiplication theorem - P(A∩ B)=P(A) P(BA) - applies universally; the cleaner product rule P(A∩ B)=P(A) P(B) follows only when the two events are independent. Exercise 13.2 makes students switch between the two forms problem after problem until the distinction is reflexive.
The exercise also forces a habit examiners reward generously: declare independence before invoking the product rule. Students who skip the declaration lose 0.5 to 1 mark even with a correct numerical answer. Recognising the four canonical setups - draws with and without replacement, tossing with two faces, multiple independent attempts, and union-complement identities - is the fastest route to a full-marks answer.
Probability Ex 13.2 Video Walkthrough
Source: Magnet Brains on YouTube
How Collegedunia's Class 12 Probability NCERT Solutions Help You Clear Exercise 13.2
The class 12 probability ncert solutions on this page address every Exercise 13.2 question in the same order as the NCERT textbook, with the multiplication-theorem step always written before the independence shortcut is invoked.
The recurring trap in Exercise 13.2 is collapsing two distinct events into one because their probabilities are equal. The Collegedunia probability class 12 solutions always state independence explicitly - either by quoting the question's framing or by verifying P(A∩ B)=P(A) P(B) numerically - before the product rule is used.
- Multiplication theorem first: every solution writes P(A∩ B)=P(A) P(BA) on its own line before switching to the independence form.
- With-vs-without replacement flagged in Q2 and Q13, where the second draw's denominator drops by one card or ball.
- Independence verified, not assumed on Q4, Q5, Q6 and Q15, with the three-condition test P(A) P(B)=P(A∩ B) shown numerically.
- De Morgan substitution shown in Q9, Q10, Q14 where "neither" or "at least one" forces a complement rewrite.
- "At least one" recipe on Q11: the complement 1-(1-p)n shortcut for n independent trials is written out so students see why it is preferred to inclusion-exclusion.
Also Check: the parallel Exercise 13.1 conditional probability solutions and the Exercise 13.3 Bayes' theorem walkthrough together form the chapter's three-exercise core. The Miscellaneous Exercise solutions bundle the trickiest mixed-method items.
Multiplication Theorem and Independence Used in Class 12 Maths Exercise 13.2
The box below states the two identities exactly as the exercise uses them. The product rule for independent events is a corollary of the multiplication theorem - never the other way round.
Independent events: events A and B are independent iff P(A∩ B)=P(A) P(B). Equivalently, P(AB)=P(A) and P(BA)=P(B).
Three pairwise + one joint: for three events to be mutually independent, all three pairwise products and the triple product must equal the corresponding joint probabilities.
"At least one" identity: for independent A1,,An, P(at least one occurs)=1-i=1n(1-P(Ai)).
In Q12 and Q13, the "at least one head/red" framing triggers the complement form directly. Q11 in particular shows the three-trial die toss where the complement 1-(1/2)3=7/8 is one line of arithmetic, while the inclusion-exclusion expansion would take six.
Independence is symmetric and pair-only by default - if a question says "A,B,C are pairwise independent" it does NOT imply they are mutually independent. CBSE has not yet asked this distinction at the 5-mark level in Class 12, but JEE Advanced sets it almost every alternate year.
Probability Class 12 Maths NCERT Solutions Exercise 13.2: Question-Wise Answer Map
The table below records the final answer for every Exercise 13.2 problem. Use it to verify your setup - a correct multiplication-theorem application almost always produces the listed value.
| Q No. | What it tests | Answer |
|---|---|---|
| 1 | Independence product rule | P(A∩ B)=325 |
| 2 | Two black cards without replacement | 25102 |
| 3 | Box-of-oranges approval | 4491 |
| 4 | Coin-head and die-3 independence check | Yes, independent |
| 5 | Even-number and red-face independence | Not independent |
| 6 | Numerical independence verification | Not independent |
| 7 | p when (i) mutually exclusive (ii) independent | (i) p=110, (ii) p=15 |
| 8 | Four probabilities from P(A),P(B) | 0.12, 0.58, 0.42, 0.28 |
| 9 | Not-A and not-B | 38 |
| 10 | Independence check from union-complement | Independent |
| 11 | Independent events: 5 probabilities | 0.18, 0.72, 0.42, 0.28, 0.4 |
| 12 | At least one odd number in three throws | 78 |
| 13 | With-replacement draws: both, first, at least one | 2581,2081,6581 |
| 14 | Problem solved independently by A and B | (i) 12 (ii) 13 |
| 15 | Card-from-deck independence cases | (i) Yes (ii) Yes (iii) No |
| 16 | Newspaper-readers conditional probabilities | (a) 12 (b) 23 (c) 13 |
| 17 | MCQ: pair-of-dice even prime | (D) 136 |
| 18 | MCQ: condition for independence | (D) P(AB)=P(A) |
The spread of denominators across the 18 answers shows how varied the multiplication-theorem applications are. A multiplication-theorem or independence question has been set in every CBSE Class 12 Maths board paper for the last five years - usually as a 3-mark short answer, occasionally as part of a 4-mark case study.
Probability Weightage Compared Across Class 12 Maths Chapters
Probability sits comfortably in the upper half of the Class 12 Maths weightage table. The chapter alone accounts for 8 of the 80 board marks, with Exercise 13.2's multiplication-theorem cluster contributing 3 to 4 of them.
| Chapter | Topic | Avg CBSE Marks |
|---|---|---|
| Ch 7 | Integrals | 10 marks |
| Ch 9 | Differential Equations | 9 marks |
| Ch 10 | Vector Algebra | 9 marks |
| Ch 13 | Probability | 8 marks |
| Ch 11 | Three Dimensional Geometry | 8 marks |
| Ch 6 | Application of Derivatives | 8 marks |
| Ch 5 | Continuity & Differentiability | 7 marks |
| Ch 12 | Linear Programming | 5 marks |
Common Mistakes Students Make in Class 12 Maths Chapter 13 Exercise 13.2
Examiner reports for the last three CBSE Class 12 Maths boards highlight the same four error patterns on the multiplication-theorem questions of Exercise 13.2.
- Mixing replacement modes in Q2 - using 2652· 2652 instead of 2652· 2551. The 52-pack drops to 51 after the first draw.
- Forgetting De Morgan on Q9 - writing P(A∩ B)=1-P(A∩ B) instead of P(A∩ B)=1-P(A∪ B).
- Expanding "at least one" by inclusion-exclusion instead of using 1-(1-p)n. The Q12 odd-number-in-three-throws problem is one line with the complement and six lines without.
- Pairwise versus mutually independent confusion - the question states "A,B,C are independent" and the student verifies only pairwise products. Mutual independence needs the triple condition too.
- Internal-inconsistency oversight in Q10: given data must satisfy P(A∩ B)≤ min(P(A),P(B)). Always do this sanity check before invoking any formula.
Exercise-wise Breakdown of the Probability Chapter
The Probability chapter splits into 3 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| Exercise | Topic Tested | Question Count |
|---|---|---|
| Exercise 13.1 | Conditional probability and the chain rule | 17 questions |
| Exercise 13.2 | Multiplication theorem and independence of events | 18 questions |
| Exercise 13.3 | Total probability and Bayes' theorem | 14 questions |
| Miscellaneous Exercise | Mixed-method probability problems | 10 questions |
NCERT Solutions for Class 12 Mathematics: All Chapters
Chapter-by-chapter NCERT Solutions for the rest of Class 12 Mathematics, each mapped to the 2026-27 print.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1 | Relations and Functions NCERT Solutions |
| Chapter 2 | Inverse Trigonometric Functions NCERT Solutions |
| Chapter 3 | Matrices NCERT Solutions |
| Chapter 4 | Determinants NCERT Solutions |
| Chapter 5 | Continuity and Differentiability NCERT Solutions |
| Chapter 6 | Application of Derivatives NCERT Solutions |
| Chapter 7 | Integrals NCERT Solutions |
| Chapter 8 | Application of Integrals NCERT Solutions |
| Chapter 9 | Differential Equations NCERT Solutions |
| Chapter 10 | Vector Algebra NCERT Solutions |
| Chapter 11 | Three Dimensional Geometry NCERT Solutions |
| Chapter 12 | Linear Programming NCERT Solutions |
PDF Download Formats and Languages for the Probability Chapter
The probability class 12 pdf on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors a topper's Sunday-revision pace | 5-7 MB |
The probability class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- NCERT-faithful: every definition and theorem in the probability class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: the probability class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: the probability class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter, including the multiplication theorem and Bayes' theorem.
- Solutions PDF separate: the probability class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: students on the Maharashtra board, HSC, or any state-board syllabus will find the same multiplication theorem in this chapter - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the probability class 12 formulas pdf), and the full probability class 12 ncert solutions on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Probability Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Probability chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the probability marks:
| Template | Typical Marks | What it tests |
|---|---|---|
| Multiplication theorem + independence check | 3 marks | Verify whether two given events are independent, then compute a joint probability. |
| Bayes' theorem case study | 4 marks | Real-world setup with two or three hypotheses; CBSE 2021+ case-study format. |
| "At least one" complement | 2-3 marks | 1-(1-p)n for n independent trials - usually a dice or coin scenario. |
Walking through one example of each template before the exam covers most of the predictable probability class 12 important questions students will see on board day. The probability class 12 exercise 13.2 cluster contributes the multiplication-theorem template directly, so revising this exercise twice closes off one of the three CBSE templates entirely. The class 12 maths probability ncert solutions on this page already worked through Q12 (at least one), Q14 (independence + product) and the Ex 13.3 Bayes' set - which together exhaust the three CBSE templates.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Probability chapter across recent CBSE Class 12 Maths boards:
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Multiplication theorem + case-study item | 5-6 marks |
| 2023 | Bayes' theorem case study + assertion-reason MCQ | 5-6 marks |
| 2022 | Independence verification + 2-mark complement | 5-7 marks |
| 2021 | Conditional probability + Bayes' theorem | 4-5 marks |
| 2020 | Without-replacement draw + 3-mark independence | 5 marks |
How Probability NCERT Solutions Pair with the Notes and the Formula Sheet
The class 12 probability solutions on this page work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| Resource | Use it for | When |
|---|---|---|
| Probability NCERT Solutions (this page) | Step-by-step worked Exercise 13.2 answers | Second pass, during NCERT practice |
| Probability Notes | Theory, definitions, exam patterns | First pass, before practice |
| Probability class 12 formulas PDF | One-page identity recall sheet | Third pass, alongside mock papers |
| NCERT Exemplar Probability Solutions | Advanced JEE-style problems | Fourth pass, JEE Main practice |
Around 60 percent of the probability vocabulary appears on all four pages, so cross-resource use reinforces recall without adding study time. The class 12 maths probability ncert solutions cover every Exercise 13.2 question; the Notes page summarises the theorems; the Formula Sheet condenses every identity to A4; the Exemplar Solutions push into JEE Main territory.
- The probability class 12 solutions for each individual exercise are indexed by exercise number in the Exercise-wise Breakdown table above.
- The probability class 12 formulas reference sheet is the same A4 file students sometimes refer to as probability class 12 all formulas.
- The ncert probability class 12 notes pair with this Exercise 13.2 page for theory-before-practice revision.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra HSC textbook PDF share the same core definitions.
- For class-first search phrasings - class 12 probability solutions, class 12 probability ncert solutions, ncert class 12 probability solutions, class 12 probability exercise 13.2 - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this exercise cleanly:
| Reference | How it maps to Exercise 13.2 |
|---|---|
| RD Sharma Class 12 Probability | Multiplication theorem covered in chapter section "Independent Events"; question patterns overlap with NCERT at ~70%. |
| ML Aggarwal Class 12 Probability | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo probability class 12 | Free online walkthroughs; useful for video-style learning. |
| Shaalaa probability class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| NCERT Exemplar Class 12 Probability | Advanced problems for JEE Main and JEE Advanced preparation. |
Student Pulse: Probability Difficulty Survey
All NCERT Solutions for Probability Ex 13.2 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 13 Probability Ex 13.2 is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.
Questions
If P(A)=35 and P(B)=15, find P(A∩ B) if A and B are independent events.
Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.
A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.
A fair coin and an unbiased die are tossed. Let A be the event ``head appears on the coin'' and B be the event ``3 on the die''. Check whether A and B are independent events or not.
A die marked 1,2,3 in red and 4,5,6 in green is tossed. Let A be the event ``the number is even'' and B be the event ``the number is red''. Are A and B independent?
Let E and F be events with P(E)=35, P(F)=310 and P(E∩ F)=15. Are E and F independent?
Given that the events A and B are such that P(A)=12, P(A∪ B)=35 and P(B)=p. Find p if they are
(i) mutually exclusive (ii) independent.
Let A and B be independent events with P(A)=0.3 and P(B)=0.4. Find
(i) P(A∩ B) (ii) P(A∪ B) (iii) P(AB) (iv) P(BA).
If A and B are two events such that P(A)=14, P(B)=12 and P(A∩ B)=18, find P(not A and not B).
Events A and B are such that P(A)=12, P(B)=712 and P(not A or not B)=14. State whether A and B are independent?
Given two independent events A and B such that P(A)=0.3, P(B)=0.6. Find
(i) P(A and B) (ii) P(A and not B) (iii) P(A or B) (iv) P(neither A nor B).
A die is tossed thrice. Find the probability of getting an odd number at least once.
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that
(i) both balls are red.
(ii) first ball is black and second is red.
(iii) one of them is black and other is red.
Probability of solving specific problem independently by A and B are 12 and 13 respectively. If both try to solve the problem independently, find the probability that
(i) the problem is solved, (ii) exactly one of them solves the problem.
One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent?
(i) E: ``the card drawn is a spade''; F: ``the card drawn is an ace''.
(ii) E: ``the card drawn is black''; F: ``the card drawn is a king''.
(iii) E: ``the card drawn is a king or queen''; F: ``the card drawn is a queen or jack''.
In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random.
(a) Find the probability that she reads neither Hindi nor English newspapers.
(b) If she reads Hindi newspaper, find the probability that she reads English newspaper.
(c) If she reads English newspaper, find the probability that she reads Hindi newspaper.
The probability of obtaining an even prime number on each die, when a pair of dice is rolled, is
(A) 0 (B) 13 (C) 112 (D) 136.
Two events A and B will be independent, if
(A) A and B are mutually exclusive
(B) P(A'B')=[1-P(A)][1-P(B)]
(C) P(A)=P(B)
(D) P(A)+P(B)=1.
Probability Class 12 NCERT Solutions - Frequently Asked Questions
Ques. How many questions are in Class 12 Maths Chapter 13 Exercise 13.2?
Ans. Exercise 13.2 of Class 12 Maths Chapter 13 Probability has 18 questions in the 2026-27 NCERT. Questions 1 to 16 are subjective items on the multiplication theorem and independence, while Questions 17 and 18 are single-correct MCQs. The probability class 12 ncert solutions ex 13.2 set on this page covers every one of them with step-by-step working.
Ques. What is the multiplication theorem on probability in Class 12 Maths Chapter 13?
Ans. The multiplication theorem states that for any two events A and B with P(A),P(B)>0, P(A∩ B)=P(A) P(BA)=P(B) P(AB). When A and B are independent, the conditional probability P(BA) reduces to P(B), giving the product rule P(A∩ B)=P(A) P(B).
Ques. What is the difference between mutually exclusive and independent events?
Ans. Mutually exclusive events cannot occur together, so P(A∩ B)=0; independent events have no influence on each other, so P(A∩ B)=P(A) P(B). Q7 of Exercise 13.2 contrasts the two explicitly - the answers differ (p=110 vs p=15) precisely because the two conditions are not the same.
Ques. How do you verify whether two events are independent in Class 12 probability?
Ans. Compute P(A), P(B) and P(A∩ B) from the sample space. If P(A)· P(B)=P(A∩ B), the events are independent; otherwise they are dependent. Q4, Q5, Q6 and Q15 of the probability class 12 ncert solutions exercise 13.2 walk through this test on dice and card examples.
Ques. Why does Q2 use 25 51 instead of 26 52 for the second draw?
Ans. Q2 specifies "without replacement", so after one black card is drawn the deck has 51 cards with 25 black left. The product becomes 2652· 2551=25102. If the question said "with replacement", the second factor would stay at 2652 and the answer would be 14.
Ques. What is the "at least one" formula used in Exercise 13.2 Q12 and Q13?
Ans. For n independent trials each with success probability p, the probability of at least one success is 1-(1-p)n. Q12 throws a die three times for an odd number (p=1/2), giving 1-(1/2)3=7/8. Q13 draws balls twice with replacement and asks for "at least one red", giving 1-(4/9)2=65/81.
Ques. How do I download the Class 12 Maths Chapter 13 Exercise 13.2 NCERT Solutions PDF?
Ans. Use the green download button on the PDF card at the top of this page to save the Collegedunia Class 12 Maths Chapter 13 Probability Exercise 13.2 NCERT Solutions PDF. The file is free, ad-free, mapped to the 2026-27 NCERT edition, and runs 22 pages with each question on its own page.
Ques. Is Probability Exercise 13.2 important for JEE Main and CUET?
Ans. Yes. The multiplication theorem and independence shortcut are tested on roughly 60-70% of JEE Main shifts that include a probability question, usually as a draw-without-replacement or "at least one" setup. CUET (UG) Mathematics pulls 1-2 MCQs per slot from the same template, often a direct lift from the NCERT Q17 / Q18 style.
Ques. How much weightage does the Probability chapter carry in CBSE Class 12 Maths?
Ans. Probability carries 8 marks out of 80 in the CBSE Class 12 Mathematics board paper, of which the multiplication-theorem and independence sub-cluster contributes 3 to 4 marks. The remaining marks come from Bayes' theorem and the case-study item in the Section E block.
Ques. What is an independent event in Class 12 Maths?
Ans. Two events A and B are independent if the occurrence of one does not change the probability of the other. Formally, P(A∩ B)=P(A) P(B), or equivalently P(AB)=P(A). The NCERT introduces this definition in Chapter 13 Section 13.4 of the 2026-27 textbook.
Ques. How is the multiplication theorem of probability defined?
Ans. The multiplication theorem of probability defines the joint probability of two events as P(A∩ B)=P(A) P(BA). It generalises to n events as P(A1∩ ∩ An)=P(A1) P(A2A1) P(A3A1∩ A2) P(AnA1∩ ∩ An-1). This is the chain rule that underlies every Exercise 13.2 problem.
Ques. What are mutually independent events?
Ans. Three or more events are mutually independent if every pairwise product and every higher-order product equals the corresponding joint probability. For three events A,B,C that means P(A∩ B)=P(A)P(B), P(B∩ C)=P(B)P(C), P(A∩ C)=P(A)P(C) AND P(A∩ B∩ C)=P(A)P(B)P(C). Pairwise independence alone is weaker - JEE Advanced regularly tests the distinction.
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