The Probability Class 12 NCERT Solutions on this page cover Exercise 13.3 of Class 12 Mathematics Chapter 13 Probability in full. Exercise 13.3 focuses on the theorem of total probability and Bayes' theorem, mapped one question per page to the 2026-27 NCERT.
- CBSE Weightage: Bayes' theorem and the theorem of total probability together carry 5 to 6 marks in the Class 12 Maths board paper, almost always as one long-answer item from this exercise.
- Skill split: total-probability set-up (Q1, Q9), classical Bayes' inversion (Q2 to Q8, Q11), tree-with-branching priors (Q10, Q12), and short MCQs (Q13, Q14).
Student Pulse - Probability Difficulty (March 2026 Collegedunia survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated Bayes' theorem as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.5 marks from confusing the prior with the likelihood in a Bayes' problem.
- 74% of JEE aspirants reported re-revising Exercise 13.3 at least twice in the week before the JEE Main exam.
- Most-skipped sub-topic: Question 5 (the blood-test base-rate problem), skipped by roughly 22% of students attempting Exercise 13.3 for the first time.
- Toppers reported that drawing the probability tree before any algebra added 1-2 marks on the Bayes' long-answer question.
Each probability class 12 ncert solutions step on this page is curated by subject experts, mapped to the 2026-27 NCERT, and refined against the last five years of CBSE Class 12 Maths, JEE Main, JEE Advanced and CUET (UG) Mathematics papers.
Probability Class 12 NCERT Solutions Exercise 13.3: Question-Wise Answer Map
The probability class 12 ncert solutions on this page tackle Exercise 13.3 in the same order as the printed NCERT 2026-27 textbook.
The exercise threads two ideas across 14 problems: first compute the total probability of an event using a partition, then invert that calculation to find the posterior probability of a "cause". The table records the final answer for each question so students can self-check after attempting.
| Q No. | What it asks | Answer |
|---|---|---|
| 1 | Urn with replacement and 2 extra balls, second draw red | 12 |
| 2 | Two bags, red drawn: posterior on first bag | 23 |
| 3 | Hosteller vs day scholar given A grade | 913 |
| 4 | Knows vs guesses, given the answer is correct | 1213 |
| 5 | Blood-test base-rate fallacy: P(disease+) | 22133 ≈ 0.165 |
| 6 | Three coins, head observed: P(two-headed) | 49 |
| 7 | Insurance: posterior on scooter driver | 152 |
| 8 | Two machines, defective: P(Bdefective) | 14 |
| 9 | Two competing groups, new product introduced | 311 |
| 10 | Die then coins, exactly one head: posterior on 1-4 | 811 |
| 11 | Three operators, defective: posterior on A | 534 |
| 12 | Lost card diamond given two diamonds drawn | 1150 |
| 13 | MCQ: A speaks truth, head reported | (A) 45 |
| 14 | MCQ: A⊂ B , conditional inequality | (C) P(AB)≥ P(A) |
Questions 2, 3, 4, 6, 8 and 11 are textbook Bayes' inversions; Q5 (the blood-test problem) and Q12 (the lost-card problem) are the most-asked items because they extend Bayes' theorem to the base-rate fallacy and to a non-trivial partition. A Bayes' theorem item has appeared in five of the last six CBSE Class 12 Maths papers, usually as a 4-mark or 5-mark long answer.
Probability Ex 13 3 Video Walkthrough
Source: Magnet Brains on YouTube
Total Probability & Bayes' Definitions Used in Class 12 Maths Exercise 13.3
These notes for Exercise 13.3 address the formulae in the same order as the printed NCERT textbook.
The box below collects every rule Exercise 13.3 uses. Total probability is the forward step (priors times likelihoods, summed across a partition), and Bayes' theorem is the inverse step (one branch divided by that sum).
Theorem of total probability: P(A)=i=1n P(Ei) P(AEi) .
Bayes' theorem: P(EiA)=P(Ei) P(AEi)j P(Ej) P(AEj) .
Prior & likelihood: P(Ei) is the prior; P(AEi) is the likelihood; P(EiA) is the posterior.
Every Bayes' problem in Exercise 13.3 has the same template: identify the partition, list the priors and likelihoods, compute the total probability of the observed event, then divide one branch by that total. A probability tree with priors on the first split and likelihoods on the second split is the fastest visual aid.
How These NCERT Solutions Help You Clear Class 12 Maths Exercise 13.3
The class 12 probability ncert solutions on this page take the printed NCERT order and add a tree diagram, an explicit prior table and a sanity check on every Bayes' answer.
The single most common error in this exercise is confusing the prior P(Ei) with the likelihood P(AEi) , or stopping at the total probability without performing the Bayes' inversion. Every probability solutions class 12 step on this page lists both the prior table and the likelihood table before any algebra, so the substitution is mechanical.
- Prior and likelihood listed separately before each Bayes' computation, so the partition is unambiguous.
- Tree diagram referenced in Q4, Q5, Q6 and Q10 where the partition has three or more branches.
- Sanity check on the posterior shown after Q5 and Q12, where the result is counter-intuitive (a positive test does not mean a high probability of disease).
- Combinatorial alternative shown in Q12, using nr selections instead of a tree.
- MCQ shortcut on Q13 and Q14 explained in two lines each, so the assertion-reason style is also covered.
Year-Wise Question Pattern for Class 12 Maths Chapter 13 Exercise 13.3
Total probability and Bayes' theorem are the highest-yield items in the Probability chapter because they consistently appear as a 4-mark or 5-mark long answer. The table maps the last five CBSE Class 12 Maths sittings, latest first.
| Year | Marks from Ex 13.3 style | Question tested |
|---|---|---|
| 2025 | 5 | Bayes' theorem on three machines producing defective items (Q11 analogue) |
| 2024 | 4 | Two-bag Bayes' inversion with red ball drawn (Q2 analogue) |
| 2023 | 5 | Insurance / accident-driver Bayes' problem (Q7 analogue) |
| 2022 | 4 | Total probability set-up across hostel and day-scholar partition (Q3) |
| 2021 | 5 | Three-coin Bayes' problem with biased coin (Q6 analogue) |
Exercise 13.3 style content has carried 4 to 5 marks in every recent CBSE Class 12 Maths paper, and the same template surfaces in JEE Main and CUET (UG) Mathematics short MCQs each year.
Related Resources for Class 12 Maths Chapter 13 Probability
Also Check: the sister-resource pages for Probability use the same notation and equation numbering as the printed NCERT 2026-27 release.
- Chapter-level NCERT Solutions for Class 12 Maths Chapter 13 Probability
- Formula Sheet for Class 12 Maths Chapter 13 Probability
- NCERT Notes for Class 12 Maths Chapter 13 Probability
- NCERT Book PDF for Class 12 Maths Chapter 13 Probability
- NCERT Exemplar Solutions for Class 12 Maths Chapter 13 Probability
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 |
Common Mistakes Students Make in Class 12 Maths Chapter 13 Exercise 13.3
The Collegedunia experts have written these probability solutions class 12 in formal mathematical notation, line by line, matching the convention of the printed NCERT.
- Swapping priors and likelihoods in the Bayes' denominator. In Q3, the prior is 0.6 (hostel proportion) and the likelihood is 0.3 (A grade given hostel); writing them the other way around drops 2 marks.
- Stopping at the total probability and forgetting to divide. Q1 needs only the total probability of red; Q2 onwards needs the Bayes' division.
- Computing P(A) for the wrong event. In Q12, the conditioning event is "two diamonds drawn from the remaining 51 cards", not "two diamonds drawn from 52".
- Forgetting that the priors must sum to 1. In Q11 the operators contribute 0.5+0.3+0.2=1 ; a missing branch fails the partition check.
- Reading the MCQ in Q14 as P(AB)≤ P(A) . Set inclusion A⊂ B gives P(AB)=P(A)/P(B)≥ P(A) , not the reverse.
NCERT Solutions for Class 12th Maths Chapter 13 Probability: All Exercises
The table below lists every other exercise of Class 12 Maths Chapter 13 Probability currently published, mapped to the 2026-27 NCERT.
| Exercise | NCERT Solutions |
|---|---|
| Exercise 13.1 | Exercise 13.1 Solutions (Conditional Probability) |
| Exercise 13.2 | Exercise 13.2 Solutions (Multiplication Theorem & Independence) |
| Miscellaneous | Miscellaneous Exercise Solutions (Mixed-Method Problems) |
The Probability chapter notes are available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
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 | Theorem of total probability and Bayes' theorem |
| Miscellaneous Exercise | Mixed probability problems |
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 ncert probability class 12 file follows the same equation numbering as the printed textbook. 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 how a topper writes the chapter under 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 about the resource:
- NCERT-faithful: Every definition, theorem and exercise on the probability class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The ncert probability class 12 file is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The probability all formulas class 12 PDF is a one-page A4 reference sheet listing every identity used in the chapter, including the Bayes' theorem statement; some students search for it as class 12 probability formulas or formula of probability class 12.
- Solutions PDF separate: The probability solutions class 12 PDF gives every NCERT exercise worked out step by step, with the tree diagram drawn for every Bayes' problem.
- State-board alignment: Students on the Maharashtra board (HSC) or any state-board syllabus will find the same definitions in this chapter - only the exercise numbers differ; the probability distribution class 12 shaalaa walkthrough mirrors the NCERT version.
Tip: Many toppers keep two parallel copies - a printed probability formulas for class 12 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 2019 to 2024 boards. Three Bayes' templates account for over 80% of the marks Exercise 13.3 contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Two-source posterior (bags / boxes) | 4 marks | Two priors, one likelihood per branch, single Bayes' inversion - Q2 and Q4 are textbook examples. |
| Three-source posterior (machines / coins) | 5 marks | Three branches, mixed likelihoods - Q6, Q8 and Q11 are the recurring shapes. |
| Base-rate / rare-disease item | 4 marks | Tiny prior, near-perfect likelihood, counter-intuitive posterior - Q5 is the canonical example. |
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.
- Probability class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The probability class 12 important questions with full solutions are reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching Probability exemplar set adds advanced Bayes' problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - usually short total-probability checks or Bayes' assertion-reason items.
Year-wise PYQ Distribution for Probability
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 | Bayes' two-source posterior + 2-mark conditional | 5-6 marks |
| 2023 | Bayes' three-source posterior + assertion-reason MCQ | 6-7 marks |
| 2022 | Total probability set-up + 3-mark independence check | 5-7 marks |
| 2021 | Bayes' three-coin item + definition recall | 5 marks |
| 2020 | Two-bag Bayes' + 2-mark conditional probability | 5 marks |
The full probability class 12 important questions with solutions set - every year, every paper, every question type - is linked from the PYQ page at the bottom of this article.
How the Probability Solutions Pair with NCERT Notes and the Formula Sheet
The class 12 probability ncert 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 solved exercises | Second pass, during NCERT practice |
| Probability Notes | Theory, definitions, exam patterns | First pass, before practice |
| Probability class 12 formulas PDF | One-page identity recall (Bayes' + total probability) | 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 resource pages, so cross-resource use reinforces recall without adding study time.
- The probability solutions class 12 on this page cover every back-of-chapter exercise plus the miscellaneous exercise.
- The probability class 12 all formula reference sheet is the same A4 file students sometimes refer to as probability all formulas class 12 or all formula of probability class 12 - it lists every identity used in the chapter, including the Bayes' denominator expansion.
- For students searching class 12 probability formulas, class 12 probability solutions or probability ncert class 12, the same file set covers the request.
- Some students look up the chapter as ch probability class 12 in their term-break revision planner; whichever phrasing is used, the file maps to the same NCERT 2026-27 chapter.
- For class-first search phrasings - class 12 probability solutions, class 12 probability ncert solutions, ncert class 12 probability solutions, ncert solutions probability class 12 - the same files cover the request.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board the chapter textbook PDF all share the same core definitions; the probability distribution class 12 shaala walkthrough mirrors Exercise 13.3 closely.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map Exercise 13.3 cleanly:
| Reference | How it maps to the NCERT |
|---|---|
| RD Sharma Class 12 Probability | Bayes' question patterns overlap with NCERT at ~70%; an advanced supplement. |
| 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. |
| Maharashtra board probability chapter textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Probability | Advanced Bayes' problems for JEE Main / JEE Advanced preparation. |
How to Use the Probability Solutions Page Most Effectively
The recommended study plan for Exercise 13.3 splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT theorem statement for total probability and Bayes'. Draw the canonical tree diagram. Mark every prior and likelihood in the textbook examples. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve textbook Example 18 and Example 19 (the two Bayes' worked examples) without looking at the solution. Compare against the printed working. Use this page if stuck. |
| Sitting 3: Exercise 13.3 | ~90 minutes | Attempt all 14 Exercise 13.3 questions. Track which ones you finish cleanly and which need a second pass. Compare against the answer map above for verification. |
For students preparing for both CBSE board and JEE Main:
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style Bayes' problem sets - sharpens speed and conceptual depth on rare-event posteriors.
- The probability class 12 important questions set on the previous-year page is the closest free analogue to a JEE Main problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall of the Bayes' denominator.
All NCERT Solutions for Probability Exercise 13.3 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 13 Probability Exercise 13.3 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
An urn contains \(5\) red and \(5\) black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, \(2\) additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?
A bag contains \(4\) red and \(4\) black balls, another bag contains \(2\) red and \(6\) black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.
Of the students in a college, it is known that \(60\%\) reside in hostel and \(40\%\) are day scholars (not residing in hostel). Previous year results report that \(30\%\) of all students who reside in hostel attain A grade and \(20\%\) of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostlier?
In answering a question on a multiple choice test, a student either knows the answer or guesses. Let \(\dfrac{3}{4}\) be the probability that he knows the answer and \(\dfrac{1}{4}\) be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability \(\dfrac{1}{4}\). What is the probability that the student knows the answer given that he answered it correctly?
A laboratory blood test is \(99\%\) effective in detecting a certain disease when it is in fact present. However, the test also yields a false positive result for \(0.5\%\) of the healthy person tested (i.e. if a healthy person is tested, then, with probability \(0.005\), the test will imply he has the disease). If \(0.1\) percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?
There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads \(75\%\) of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?
An insurance company insured \(2000\) scooter drivers, \(4000\) car drivers and \(6000\) truck drivers. The probability of an accident are \(0.01\), \(0.03\) and \(0.15\) respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?
A factory has two machines \(A\) and \(B\). Past record shows that machine \(A\) produced \(60\%\) of the items of output and machine \(B\) produced \(40\%\) of the items. Further, \(2\%\) of the items produced by machine \(A\) and \(1\%\) produced by machine \(B\) were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine \(B\)?
Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are \(0.6\) and \(0.4\) respectively. Further, if the first group wins, the probability of introducing a new product is \(0.7\) and the corresponding probability is \(0.3\) if the second group wins. Find the probability that the new product introduced was by the second group.
Suppose a girl throws a die. If she gets a \(5\) or \(6\), she tosses a coin three times and notes the number of heads. If she gets \(1,2,3\) or \(4\), she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw \(1,2,3\) or \(4\) with the die?
A manufacturer has three machine operators \(A\), \(B\) and \(C\). The first operator \(A\) produces \(1\%\) defective items, whereas the other two operators \(B\) and \(C\) produce \(5\%\) and \(7\%\) defective items respectively. \(A\) is on the job for \(50\%\) of the time, \(B\) is on the job for \(30\%\) of the time and \(C\) is on the job for \(20\%\) of the time. A defective item is produced, what is the probability that it was produced by \(A\)?
A card from a pack of \(52\) cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.
Probability that \(A\) speaks truth is \(\dfrac{4}{5}\). A coin is tossed. \(A\) reports that a head appears. The probability that actually there was head is
(A) \(\dfrac{4}{5}\) (B) \(\dfrac{1}{2}\) (C) \(\dfrac{1}{5}\) (D) \(\dfrac{2}{5}\).
If \(A\) and \(B\) are two events such that \(A\subset B\) and \(P(B)\ne 0\), then which of the following is correct?
(A) \(P(A\mid B)=\dfrac{P(B)}{P(A)}\) (B) \(P(A\mid B)
Student Pulse - Probability Difficulty
Probability Class 12 NCERT Solutions Exercise 13.3 - Frequently Asked Questions
Ques. How many questions are in Class 12 Maths Chapter 13 Exercise 13.3?
Ans. Exercise 13.3 of Class 12 Maths Chapter 13 Probability has 14 questions in the 2026-27 NCERT. Questions 1 to 12 are word problems on the theorem of total probability and Bayes' theorem, while Questions 13 and 14 are single-correct MCQs.
Ques. What does Bayes' theorem state in Class 12 Maths Chapter 13?
Ans. Bayes' theorem says that for a partition E1,E2,,En of the sample space and an event A with P(A)≠ 0 , P(EiA)=P(Ei) P(AEi)j P(Ej) P(AEj) . The numerator is the joint probability of Ei and A ; the denominator is the total probability of A .
Ques. What is the theorem of total probability in Class 12 Maths Exercise 13.3?
Ans. The theorem of total probability states that if E1,E2,,En partitions the sample space, then P(A)=i P(Ei) P(AEi) . It is the forward computation that the Bayes' denominator uses.
Ques. How do I solve Question 5 (the blood-test problem) in Probability Class 12 Exercise 13.3?
Ans. Use Bayes' theorem with prior P(disease)=0.001 and likelihood P(+disease)=0.99 . The total probability P(+)=0.001× 0.99+0.999× 0.005=0.005985 . So P(disease+)=0.000990.005985=22133≈ 0.165 . The small prior controls the answer - a positive test is not the same as having the disease.
Ques. Which questions in Probability Class 12 Exercise 13.3 are most likely to appear in the CBSE board?
Ans. Questions 2, 4, 6, 8 and 11 (the two-source and three-source Bayes' problems) have appeared in CBSE Class 12 Maths boards every year between 2020 and 2025 in one form or another. Q5 (base-rate / rare-disease) and Q12 (lost-card) appear less often but reward 5-mark long-answer marks.
Ques. How do I download the Class 12 Maths Chapter 13 Exercise 13.3 NCERT Solutions PDF?
Ans. Use the green download button on the probability class 12 solutions card at the top of this page to save the class 12 probability ncert solutions PDF for Exercise 13.3. The file is free, ad-free and aligned to the 2026-27 NCERT edition; the same file is also available as the probability class 12 solutions pdf in HD format. The probability class 12 pdf opens directly on phone or laptop without any sign-in step.
Ques. What is the difference between class 12 probability exercise 13.1, probability class 12 exercise 13.2 and Exercise 13.3?
Ans. The probability class 12 exercise 13.1 set covers conditional probability P(AB) . The probability class 12 exercise 13.2 set covers the multiplication theorem and independence of events. Exercise 13.3 covers the theorem of total probability and Bayes' theorem - the inverse-conditional setting where students compute P(EiA) . For a year-by-year question map across all three, see the related-resources block above.
Ques. How is the probability formulas pdf for Bayes' theorem written?
Ans. The probability formulas class 12 pdf states Bayes' theorem in two equivalent forms: P(EiA)=P(Ei)P(AEi)P(A) , where P(A)=j P(Ej)P(AEj) . The probability class 12 all formulas sheet (also indexed as probability formulas class 12 pdf or probability class 12 formulas pdf) labels the prior, likelihood and posterior next to the formula so the substitution is mechanical.
Ques. What is conditional probability in Class 12 Maths?
Ans. Conditional probability P(AB) is the probability of event A given that event B has occurred, defined by P(AB)=P(A∩ B)P(B) whenever P(B)>0 . It is the building block for both the multiplication theorem (Exercise 13.2) and Bayes' theorem (Exercise 13.3).
Ques. How is the prior defined in Bayes' theorem?
Ans. The prior P(Ei) is the probability of the partition event Ei before any data is observed - in Q6 it is the 1/3 probability of picking each of three coins. The posterior P(EiA) updates this prior after observing the event A (here, "head shows"); the update factor is the likelihood P(AEi) divided by the total probability P(A) .
Ques. What are partitions of the sample space?
Ans. Partitions of the sample space S are a finite collection E1,E2,,En of pairwise disjoint events whose union is S . Every Bayes' problem in Exercise 13.3 begins by identifying a partition: in Q3 the two bags, in Q6 the three coins, in Q11 the three operators. Without a valid partition, the denominator of Bayes' theorem is undefined.
Ques. Is probability distribution deleted from Class 12 in the 2026-27 NCERT?
Ans. The earlier "probability distribution of random variables" sub-topic was rationalised out of the Class 12 NCERT in the 2023-24 print and remains out of the 2026-27 syllabus. Exercise 13.3 covers Bayes' theorem and total probability only - the random-variable section that some probability distribution class 12 shaalaa sites still list is no longer part of the CBSE Class 12 board paper.
Ques. Where can I get the class 12 probability pdf in HD?
Ans. The HD class 12 probability pdf is the same file as the probability class 12 ncert pdf linked at the top of this page, rendered at 300dpi for print. Students who use ncert solutions probability class 12 or ncert class 12 probability solutions as their search term land on the same download.
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