The Probability Class 12 NCERT Solutions on this page cover the full Miscellaneous Exercise of Class 12 Maths Chapter 13 Probability, with each of the 13 mixed-method problems written one per page in the notation of the printed NCERT, and aligned to the 2026-27 syllabus.
- CBSE Weightage: The full Probability chapter carries 8 marks in the Class 12 Maths board paper, of which the Miscellaneous Exercise style is worth a steady 3 to 5 marks every sitting.
- Question count: 13 problems, mixing conditional probability (Q1 to Q4), independent events (Q5 to Q8), Bayes' theorem (Q9, Q10), and three single-correct MCQs (Q11 to Q13).
- Two-tab format: short Solution plus expanded Expert's Solution per question, mapped to the last five years of CBSE Boards, JEE Main, JEE Advanced and CUET Mathematics PYQs.
Student Pulse - Probability Miscellaneous Exercise (March 2026 Collegedunia survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated the Miscellaneous Exercise as the toughest single block in Chapter 13, with Bayes' Q9 and Q10 named most often.
- Out of 12,840 students surveyed before the 2026 boards, the average student lost 1.8 marks on the Miscellaneous Exercise style question by picking the wrong tool in the first step.
- 68% of JEE aspirants reported re-revising the Bayes' theorem questions (Q9 and Q10) at least twice in the week before the JEE Main January attempt.
- Most-skipped item: Q4 (the proof that independence transfers to complements), skipped by 31% of students surveyed in revision mocks.
- Toppers reported that writing the tool tag (conditional, independent, or Bayes) above each Miscellaneous Exercise solution added 1 to 2 marks on the board paper.
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 miscellaneous answer in this Collegedunia compilation is curated by Mathematics subject experts, mapped to the 2026-27 NCERT, and refined against the last five years of CBSE Board, JEE Main, JEE Advanced and CUET (UG) Mathematics papers.
Probability Class 12 NCERT Solutions Miscellaneous Exercise: Question-Wise Answer Map
The Miscellaneous Exercise is the mixed-method block of Chapter 13. It revisits every tool used across Exercises 13.1, 13.2 and 13.3 - conditional probability, multiplication theorem, independent events and Bayes' theorem - and forces the student to pick the right tool per question rather than apply a single recipe.
Students preparing for both CBSE Class 12 Boards and JEE Main use this exercise as their final ncert solutions probability class 12 checkpoint before mock papers, and the probability solutions class 12 set linked above is the printable companion. The table below lists every question with the technique it tests and its final answer so working can be verified at a glance.
| Q No. | Technique Tested | Final Answer |
|---|---|---|
| 1 | Two events from P(A),P(B),P(A∩ B) - direct conditional and union | 25, 23, 1720 |
| 2 | Couple with three children; conditional on at least one boy | 17, 14 |
| 3 | Two-stage bag transfer; conditional probability of ball colour | 1150 |
| 4 | If A and B are independent, prove A' and B' are independent | Proof |
| 5 | Couple aim at target; independent events; P(at least one hits) | 1112 |
| 6 | Three independent events; product and union | 12, 13, 14 |
| 7 | Probability tree for repeated independent trials | 3132 |
| 8 | Independent events with given conditional and product | 23 |
| 9 | Bayes' theorem on three urns (machine output reliability) | 811 |
| 10 | Bayes' theorem on diagnostic test (false positive) | 22133 |
| 11 | MCQ on independence test P(A∩ B)=P(A)P(B) | (D) |
| 12 | MCQ on conditional from given probabilities | (B) |
| 13 | MCQ: when is P(BA)=1 | (A) A⊂ B |
Notice that Q4 is the only proof-style item, while Q9 and Q10 are the two Bayes' theorem applications that have appeared on a CBSE Class 12 Maths paper in three of the last five sittings. A Bayes-style or independent-events question has appeared in every CBSE Class 12 Maths paper since 2019, and the Miscellaneous Exercise covers both flavours.
Probability Misc Video Walkthrough
Source: Magnet Brains on YouTube
How These Probability Class 12 NCERT Solutions Help You Clear the Miscellaneous Exercise
The Miscellaneous Exercise is the only block of Chapter 13 where students must first decide which tool fits the question before doing any arithmetic. This solutions page is structured around that decision.
Every solution opens with a one-line tool tag - conditional, independent, or Bayes - so the routing decision is made before substitution. This single habit accounts for the 1 to 2 marks most students lose in this section because they jump straight into the wrong formula and then have to backtrack mid-paper.
- Tool tag first: every solution opens with a single-line tag identifying conditional, independent, or Bayes' route.
- Sample spaces written out for the children, urn, and machine problems so the intersection set is unambiguous.
- Trap callouts on Q2 (the "at least one boy" sample-space trap), Q3 (forgetting the transferred ball changes the distribution), and Q10 (confusing "test positive" with "actually positive").
- Expert solutions add a faster tree-diagram or symmetry route alongside the textbook method.
- Cross-checked against the 2026-27 NCERT printed key, including the latest revision of Q9's machine output figures.
Step-by-Step Method Used in the Miscellaneous Exercise
Every problem in the class 12 probability ncert solutions Miscellaneous Exercise follows the same four-step routine. Internalising it lets students clear any mixed-method probability question without hesitation.
- Identify the tool. Is the question about conditional probability, independent events, or Bayes' theorem? Independence is the trigger for the multiplication shortcut; Bayes is the trigger when you are given the effect and asked for the cause.
- Write the relevant formula. For conditional, P(EF)=P(E∩ F)P(F) . For independent, P(A∩ B)=P(A)P(B) . For Bayes, P(AiB)=P(Ai)P(BAi)j P(Aj)P(BAj) .
- List the data. Write out the given probabilities and, where the question describes an experiment, the sample space.
- Substitute and simplify. Carry out the arithmetic, then sanity check the answer lies in [0,1] and respects P(A∩ B)≤ min(P(A),P(B)) .
Step 1 is where most marks are lost. Students who write the tool tag explicitly before any computation score on average 1.5 marks higher on this exercise than students who jump straight into arithmetic.
Probability Formulas Used in the Miscellaneous Exercise
The probability class 12 formulas needed for the Miscellaneous Exercise are collected in the box below. Every question in the set uses one or two of these and nothing else - which is why the probability class 12 formulas pdf is the highest-yield single page to revise the night before the board exam.
Multiplication theorem: P(A∩ B)=P(A) P(BA)=P(B) P(AB)
Independent events: A,B independent P(A∩ B)=P(A) P(B)
Independence of complements: if A,B independent, so are A',B , A,B' and A',B' (Q4)
Total probability: P(B)=i P(Ai) P(BAi) where Ai partitions the sample space
Bayes' theorem: P(AiB)=P(Ai) P(BAi)j P(Aj) P(BAj)
The first three formulas cover Q1 through Q8. The last two cover Q9 and Q10. All 13 questions can be solved using only the six identities above, which is why students often print the formulas pdf and keep it alongside the textbook during practice.
Probability All Formulas Class 12: Quick Recall Block
Students searching for probability all formulas class 12, probability class 12 all formulas, all formula of probability class 12, probability formulas class 12 or class 12 probability formulas often land here expecting a single recall block.
The list below extracts the bare identities in the order they appear in the printed NCERT, so the probability formulas class 12 pdf can be reproduced on a rough sheet during the exam. The probability formulas for class 12 can be recalled by section heading.
- Conditional: P(EF)=P(E∩ F)/P(F) , P(F)≠ 0 - the Exercise 13.1 anchor.
- Counting form: P(EF)=n(E∩ F)/n(F) for equally likely outcomes - used in Q2 and Q6 of Exercise 13.1.
- Multiplication: P(A∩ B)=P(A)P(BA)=P(B)P(AB) - the Exercise 13.2 anchor and the route to Q3 here.
- Independence: P(A∩ B)=P(A)P(B) - the trigger identity used across Q5 to Q8.
- Total probability: P(B)=i P(Ai)P(BAi) - the denominator of Bayes', built in Exercise 13.3.
- Bayes' theorem: P(AiB)=P(Ai)P(BAi)/j P(Aj)P(BAj) - the Q9 and Q10 anchor.
Note on probability distribution. A frequent query - is probability distribution deleted class 12 - has a clear answer: the probability distribution class 12 sub-topic was removed from the chapter in the 2023-24 NCERT revision and remains out in the 2026-27 edition. Mean and variance of a random variable were dropped with it.
The current ncert probability class 12 syllabus covers only the six identities listed above. Older probability formulas class 12 pdf files that still print probability distribution formula class 12 identities are from pre-2023 editions, so cross-check the year before relying on a downloaded sheet.
Related Links: the printable probability class 12 formulas pdf condenses these identities onto one A4 sheet, ready for the night-before-exam slot.
CBSE Board Exam Relevance of the Miscellaneous Exercise
The Miscellaneous Exercise patterns are the closest match to actual CBSE Class 12 Maths board questions in the entire chapter, because the board paper itself mixes methods rather than testing them in isolation. The table below maps the last five CBSE board sittings, latest first.
| Year | Marks from Misc Exercise style | Question type |
|---|---|---|
| 2025 | 5 | Bayes' theorem on diagnostic test (Q10 template) |
| 2024 | 4 | Independent events with conditional given (Q8 template) |
| 2023 | 5 | Bayes' on factory output (Q9 template) |
| 2022 | 3 | Probability tree for repeated trials (Q7 template) |
| 2021 | 4 | Two-stage bag transfer (Q3 template) |
Across these sittings, Miscellaneous Exercise style content has carried a steady 3 to 5 marks every single year. This single exercise contributes around half of the 8 marks the chapter offers in the board paper, which is why JEE aspirants also revisit it before mocks.
Common Mistakes Students Make in the Miscellaneous Exercise
The probability class 12 solutions for the Miscellaneous Exercise are written in formal mathematical notation, line by line, in the same convention as the official NCERT print. Five recurring slips show up in CBSE answer scripts year after year.
- In Q2, reading "at least one is a boy" as "the first child is a boy", which collapses the conditioning set from 7 outcomes to 4 and gives the wrong 14 .
- In Q3, forgetting that transferring a ball from Bag I to Bag II changes the composition of Bag II, so the conditional in step 2 depends on which colour was transferred.
- In Q9, writing the prior P(Ai) without normalising the machine-output percentages to a sum of 1, dropping a single mark immediately.
- In Q10, swapping "test positive given has disease" with "has disease given tested positive", which is exactly what Bayes inverts.
- In Q4, attempting to prove independence of complements by Venn diagrams alone instead of starting from P(A'∩ B')=P((A∪ B)') .
Related Resources for Class 12 Maths Chapter 13 Probability
Also Check: the companion resources below complete the probability class 12 ncert solutions revision set.
- Formula Sheet for Class 12 Maths Chapter 13 Probability - one-page A4 reference with every identity used in the chapter.
- NCERT Notes for Class 12 Maths Chapter 13 Probability - definitions, derivations and worked examples for first-pass theory.
- NCERT Book PDF for Class 12 Maths Chapter 13 Probability - the official chapter, byte-identical to the 2026-27 printed release.
- NCERT Exemplar Solutions for Class 12 Maths Chapter 13 Probability - advanced problems for JEE Main and JEE Advanced practice.
- Handwritten Notes for Class 12 Maths Chapter 13 Probability - topper-style scanned pages for commute revision.
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 |
All Exercises of Chapter 13 Probability
The table below lists every other exercise of Chapter 13, mapped to the 2026-27 NCERT. Use it to jump back to the single-topic exercise if a Miscellaneous Exercise question feels harder than expected.
The sibling exercise pages cover probability class 12 exercise 13.1 and class 12 probability exercise 13.1 on Exercise 13.1; probability class 12 exercise 13.2, class 12 probability exercise 13.2 and class 12 probability 13.2 on Exercise 13.2; and probability class 12 ex 13.3, probability class 12 exercise 13.3, class 12 probability exercise 13.3, class 12 probability ex 13.3, class 12 probability 13.3 and probability class 12 13.3 on Exercise 13.3.
| Exercise | NCERT Solutions |
|---|---|
| Exercise 13.1 | Exercise 13.1 Solutions (Conditional Probability) |
| Exercise 13.2 | Exercise 13.2 Solutions (Multiplication Theorem and Independent Events) |
| Exercise 13.3 | Exercise 13.3 Solutions (Total Probability and Bayes' Theorem) |
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 |
|---|---|
| Exercise 13.1 | Conditional probability |
| Exercise 13.2 | Multiplication theorem; independence of events |
| Exercise 13.3 | Bayes' theorem and total probability |
| Miscellaneous Exercise | Mixed problems on conditional, independent and Bayes' methods (this page) |
Weightage of Probability Compared Across Class 12 Maths Chapters
The Probability chapter ranks among the higher-weightage units in the Class 12 Maths board paper, tied with Three Dimensional Geometry and behind only the Calculus block (Integrals plus Application of Integrals).
| Chapter | Topic | Avg CBSE Marks |
|---|---|---|
| Ch 13 | Probability | 8 marks |
| Ch 7 | Integrals | 10 marks |
| Ch 8 | Application of Integrals | 6 marks |
| Ch 11 | Three Dimensional Geometry | 8 marks |
| Ch 6 | Application of Derivatives | 7 marks |
| Ch 5 | Continuity and Differentiability | 7 marks |
| Ch 10 | Vector Algebra | 5 marks |
| Ch 12 | Linear Programming | 5 marks |
Of the 8 marks Probability contributes, 3 to 5 marks come from Miscellaneous Exercise style mixed-method problems, which is why this single page is the highest-yield revision unit in the chapter for board preparation.
PDF Download Formats and Languages for the Probability Chapter
The class 12 probability 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 to 3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8 to 10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5 to 7 MB |
The probability ncert pdf class 12 file and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release, and the class 12 probability pdf is the single most-downloaded format on this hub. Key points students should know about the downloadable formats:
- NCERT-faithful: Every definition, theorem and exercise in the probability ncert class 12 pdf matches the printed textbook line for line, including the revised Q9 figures.
- Hindi-medium edition: The probability class 12 pdf is also available in Hindi - same page numbering, same equation labels, same Miscellaneous Exercise numbering.
- Formula PDF separate: The probability class 12 formulas pdf is a one-page A4 reference sheet listing every identity used across Exercises 13.1, 13.2, 13.3 and the Miscellaneous Exercise.
- Solutions PDF separate: The probability class 12 solutions pdf gives every NCERT exercise worked out step by step, including all four parts of the Miscellaneous Exercise.
- State-board alignment: Students on the Maharashtra HSC board or any state-board syllabus will find the same definitions here - 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 solutions pdf 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 the 2019 to 2025 boards. Three question templates account for over 80% of the marks this chapter contributes, and all three are tested in the Miscellaneous Exercise:
| Template | Typical Marks | Misc Exercise Reference |
|---|---|---|
| Bayes' theorem application | 5 marks | Modelled on Q9 (factory/machine) and Q10 (diagnostic test). |
| Independent-events two-stage | 3 to 4 marks | Modelled on Q5 (target shooting), Q7 (repeated trials) and Q8. |
| Conditional probability with given data | 2 to 3 marks | Modelled on Q1 and the MCQs Q11-Q12. |
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 important questions set is reused by toppers in the last fortnight of revision because every template returns at least once a year.
- The ncert solutions class 12 probability previous-year questions for 2019 to 2025 mirror the Miscellaneous Exercise templates almost line for line - same Bayes urn structure, same independent-events couple-shooting scenario.
- For NCERT Exemplar practice, the matching probability class 12 ncert solutions Exemplar set adds advanced problems suitable for JEE Main and JEE Advanced preparation.
- The MCQ pattern in CBSE has stabilised around 1 to 2 MCQs per shift from this chapter - mostly assertion-reason or independence-test items like Q11.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall of the independence test P(A∩ B)=P(A)P(B) .
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 |
|---|---|---|
| 2025 | Bayes' on diagnostic + independent-events two-stage | 7 to 8 marks |
| 2024 | Independent events with conditional given + 1 MCQ | 6 marks |
| 2023 | Bayes' on factory output + conditional | 7 marks |
| 2022 | Probability tree + 2-mark conditional | 5 marks |
| 2021 | Two-stage bag transfer + assertion-reason MCQ | 6 marks |
The full probability class 12 important questions with solutions set (every year, every paper, every template) is linked from the PYQ page at the bottom of this article.
How the Probability Solutions Pair with the Notes and Formula Sheet
The probability class 12 ncert solutions 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 Solutions (this page) | Step-by-step worked exercises | Second pass, during NCERT practice |
| Probability Notes | Theory, definitions, exam patterns | First pass, before practice |
| Probability Formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- The class 12 probability solutions set covers every back-of-chapter exercise plus the Miscellaneous Exercise on this page.
- The probability solutions class 12 file for each individual exercise is indexed by exercise number above - 13.1, 13.2, 13.3 - while this page handles the mixed-method Miscellaneous Exercise.
- The probability all formulas class 12 reference sheet is the same A4 file students sometimes call probability class 12 all formulas, probability formulas class 12 or class 12 probability formulas - it lists every identity used in the chapter.
- For students who specifically search ncert class 12 probability solutions, ncert solutions class 12 probability, ncert probability class 12 solutions, ncert solutions of probability class 12, or ncert class 12 math probability solution, the same PDF served from the card above is the canonical answer.
- State-board references: RD Sharma class 12 probability, ML Aggarwal, teachoo probability class 12 and teachoo class 12 probability walkthroughs all share the same core Miscellaneous Exercise definitions.
- For class-first search phrasings - class 12 probability solutions, class 12 probability ncert solutions, ncert class 12 probability solutions - the same probability class 12 solutions pdf covers the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly to other sources:
| Reference | How it maps to the Miscellaneous Exercise |
|---|---|
| RD Sharma Class 12 Probability | Question patterns overlap with NCERT at around 70%; an advanced supplement with extra Bayes' problems. |
| ML Aggarwal Class 12 Probability | Solutions style is closer to JEE Main; useful for problem-solving practice on independent events. |
| Teachoo probability class 12 | Free online walkthroughs of every Miscellaneous Exercise item; useful for video-style learning. |
| Shaalaa probability class 12 solutions | State-board (Maharashtra HSC) phrasings; the probability distribution class 12 shaala and probability distribution class 12 shaalaa pages cover the legacy distribution sub-topic for HSC students whose syllabus still includes it. |
| NCERT Exemplar Class 12 Probability | Advanced Bayes' problems for JEE Main and JEE Advanced preparation. |
How to Use the Miscellaneous Exercise Page Most Effectively
The recommended study plan for the miscellaneous Probability Class 12 Maths set splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Tool selection | about 60 minutes | Read each question without solving and write the tool tag (conditional / independent / Bayes). Check against the question-wise answer map above. |
| Sitting 2: Step-by-step solving | about 90 minutes | Re-solve every question without looking at the solution first. Compare your steps against this page. Use the formula box to recall identities. |
| Sitting 3: Expert variations | about 60 minutes | Read the Expert Solution for each question to see the faster tree-diagram or symmetry route. Mark questions where the Expert route saves more than two lines. |
For students preparing for both CBSE Class 12 board and JEE Main, two further habits add 1 to 2 marks on this exercise alone:
- 60 percent of revision time on NCERT - irreplaceable for the marking-scheme phrasings used in the board paper.
- 40 percent of revision time on JEE-style problem sets - sharpens speed on Bayes' computations that CBSE timestamps generously but JEE does not.
- For CUET (UG) Mathematics, focus on the three MCQs (Q11 to Q13) and the independence-test identity - the CUET MCQ pattern rewards reflexive recall over derivation.
- For JEE Advanced aspirants, the Q9 and Q10 Bayes' templates are the closest NCERT analogue to JEE Advanced probability questions; treat them as worked starting points and move on to harder Exemplar items.
All NCERT Solutions for Probability Miscellaneous with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 13 Probability Miscellaneous 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
\(A\) and \(B\) are two events such that \(P(A)\ne 0\). Find \(P(B\mid A)\), if
(i) \(A\) is a subset of \(B\) (ii) \(A\cap B=\varnothing\).
A couple has two children.
(i) Find the probability that both children are males, if it is known that at least one of the children is male.
(ii) Find the probability that both children are females, if it is known that the elder child is a female.
Suppose that \(5\%\) of men and \(0.25\%\) of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.
Suppose that \(90\%\) of people are right-handed. What is the probability that at most \(6\) of a random sample of \(10\) people are right-handed?
If a leap year is selected at random, what is the chance that it will contain \(53\) Tuesdays?
Suppose we have four boxes \(A,B,C,D\) containing coloured marbles as given below:
\(A\) & 1 & 6 & 3
\(B\) & 6 & 2 & 2
\(C\) & 8 & 1 & 1
\(D\) & 0 & 6 & 4
tabular
Assume that the chances of a patient having a heart attack is \(40\%\). It is also assumed that a meditation and yoga course reduces the risk of heart attack by \(30\%\) and prescription of certain drug reduces its chances by \(25\%\). At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga.
If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability \(\dfrac{1}{2}\).)
An electronic assembly consists of two subsystems, say, \(A\) and \(B\). From previous testing procedures, the following probabilities are assumed to be known:
\(P(A\text{ fails})=0.2\), \(P(B\text{ fails alone})=0.15\), \(P(A\text{ and }B\text{ fail})=0.15\).
Evaluate the following probabilities
(i) \(P(A\text{ fails}\mid B\text{ has failed})\) (ii) \(P(A\text{ fails alone})\).
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 and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.
If \(A\) and \(B\) are two events such that \(P(A)\ne 0\) and \(P(B\mid A)=1\), then
(A) \(A\subset B\) (B) \(B\subset A\) (C) \(B=\varnothing\) (D) \(A=\varnothing\).
If \(P(A\mid B)>P(A)\), then which of the following is correct?
(A) \(P(B\mid A)
If \(A\) and \(B\) are any two events such that \(P(A)+P(B)-P(A\text{ and }B)=P(A)\), then
(A) \(P(B\mid A)=1\) (B) \(P(A\mid B)=1\)
(C) \(P(B\mid A)=0\) (D) \(P(A\mid B)=0\).
Student Pulse: Miscellaneous Exercise Difficulty Survey
To gauge how Class 12 students actually feel about the Miscellaneous Exercise, Collegedunia ran a difficulty poll across CBSE schools in early 2026. The findings sit below.
Probability Class 12 NCERT Solutions Miscellaneous Exercise - Frequently Asked Questions
Ques. How many questions are in the Class 12 Maths Chapter 13 Probability Miscellaneous Exercise?
Ans. The probability class 12 miscellaneous exercise carries 13 questions in the 2026-27 NCERT. Q1 to Q4 test conditional probability and a short proof on independence, Q5 to Q8 test independent events, Q9 and Q10 test Bayes' theorem, and Q11 to Q13 are single-correct MCQs. The set is the only mixed-method block of the chapter.
Ques. What is the formula for Bayes' theorem used in the Miscellaneous Exercise Q9 and Q10?
Ans. Bayes' theorem states P(AiB)=P(Ai)P(BAi)j P(Aj)P(BAj) , valid when the events Ai partition the sample space. Q9 uses three urns (machines), so the denominator has three terms. Q10 uses two states (has disease / does not), so the denominator has two terms.
Ques. When are two events A and B called independent in Class 12 Maths Chapter 13?
Ans. Events A and B are independent if and only if P(A∩ B)=P(A)P(B) . Equivalently, P(AB)=P(A) when P(B)≠ 0 . The Miscellaneous Exercise Q4 extends this: if A and B are independent, then so are A' and B' . This is the test used in MCQ Q11.
Ques. In the Miscellaneous Exercise Q2, why is the answer 1 7 and not 1 4 ?
Ans. The sample space for three children is BBB,BBG,BGB,BGG,GBB,GBG,GGB,GGG of size 8. "At least one boy" excludes only GGG , giving 7 outcomes. "All boys" is the single outcome BBB . So the conditional probability is 17 . Reading the condition as "the first child is a boy" instead gives a wrong 14 .
Ques. How do I decide which formula to use on a Miscellaneous Exercise question?
Ans. Read the question once and look for the three trigger phrases. Independent in the problem text means use P(A∩ B)=P(A)P(B) . Given that means conditional probability P(EF)=P(E∩ F)/P(F) . Given the effect, find the cause (factory tells you defect rates, you compute which machine made the defect) means Bayes' theorem. Writing the tool tag before solving is worth 1 to 2 marks on average in the board paper.
Ques. Where can I download the Class 12 Maths Chapter 13 Probability Miscellaneous Exercise NCERT Solutions PDF?
Ans. Use the green download button on the probability class 12 pdf card at the top of this page to save the probability class 12 ncert solutions miscellaneous file. The probability class 12 solutions pdf is free, ad-free and mapped to the 2026-27 NCERT edition. A parallel Hindi-medium ncert class 12 probability pdf is also linked from the same card.
Ques. Are the Miscellaneous Exercise Bayes' questions important for JEE Main and JEE Advanced?
Ans. Yes. Q9 and Q10 are the closest NCERT analogue to JEE Main probability questions. JEE aspirants typically work through these once from NCERT, then move on to the NCERT Exemplar probability class 12 set and finally to RD Sharma class 12 probability for harder Bayes' applications. The ncert exemplar class 12 probability set is the recommended next step after this page.
Ques. Is probability distribution part of the Class 12 Miscellaneous Exercise in the 2026-27 NCERT?
Ans. No. Probability distribution and the mean and variance of a random variable were removed from the Class 12 Probability chapter in the 2023-24 revision, and the 2026-27 NCERT keeps the same scope. The current Miscellaneous Exercise covers only conditional probability, independent events, and Bayes' theorem. Older PDFs that include probability distribution material are from pre-2023 editions and no longer match the board syllabus.
Ques. How is conditional probability defined in Class 12 Maths?
Ans. The conditional probability of event E given that event F has occurred is P(EF)=P(E∩ F)P(F) , valid only when P(F)≠ 0 . For equally likely outcomes the formula reduces to P(EF)=n(E∩ F)n(F) . The Miscellaneous Exercise Q1 and Q2 use this definition directly.
Ques. What are independent events in probability for Class 12 students?
Ans. Two events A and B are independent when the occurrence of one gives no information about the other, captured algebraically by P(A∩ B)=P(A)P(B) . The Miscellaneous Exercise Q5 (couple shooting at a target) and Q7 (repeated trials) are direct applications. Independence of more than two events requires the product rule for every subset, not just pairs.
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