The Probability Formula given here cover Class 12 Mathematics Chapter 13 Probability in a compact, recall-friendly format. The Probability Formula are split across the standard NCERT sections of conditional probability, Bayes' theorem and Bernoulli trials, with each formula in the formula sheet presented with its assumptions explicitly stated.
- CBSE Weightage: 8 to 10 marks in the Class 12 Maths board paper, typically a 5-mark long answer on Bayes or the binomial distribution plus a 3-mark case-study on conditional probability.
The chapter holds 10 working formulas across 5 concept blocks (conditional, multiplication, total probability, Bayes, binomial distribution), and this Collegedunia sheet places all of them on a single printable PDF with the symbol meanings spelt out.
The reference below is grouped by concept block so you can drill one section at a time.
This Formula Sheet is curated by Class 12 Maths experts at Collegedunia, mapped to the 2026-27 NCERT edition and the last five years of CBSE Board and JEE Main papers.
Probability All Formulas Class 12 at a Glance
Chapter 13 is the final and highest-scoring chapter of the Class 12 Maths syllabus for most students who attempt it fully. About 80% of CBSE board marks on the Probability Formula are awarded for correctly applying the conditional, total-probability, and Bayes formulas in sequence.
The 2026-27 NCERT print keeps the Probability Formula focused on five blocks: classical definition, conditional probability, multiplication rule, total probability with Bayes, and the binomial distribution.
Probability Video Walkthrough
Source: Magnet Brains on YouTube
Classical Definition and the Addition Rule of Probability
The first three formulas form the foundation for the rest of the Probability Formula. Every later result, including Bayes and the binomial distribution, is built on the addition rule and the classical definition below.
| # | Concept | Formula |
|---|---|---|
| 1 | Classical (a priori) probability | P(A) = number of favourable outcomestotal number of outcomes |
| 2 | Addition rule for two events | P(A ∪ B) = P(A) + P(B) - P(A ∩ B) |
| 3 | Mutually exclusive events | P(A ∪ B) = P(A) + P(B) , since P(A ∩ B) = 0 |
The classical definition assumes all outcomes are equally likely (fair coins, fair dice, well-shuffled cards). When they are not, fall back to the axiomatic definition the NCERT introduces in Section 13.1.
Conditional Probability and the Multiplication Theorem
The Probability Formula address this in the same order as the NCERT textbook.
Conditional probability is the doorway to the rest of the chapter. CBSE markers award step marks if you write the definition before substituting numbers, so anchor this formula first.
| # | Concept | Formula |
|---|---|---|
| 4 | Conditional probability of A given B | P(A B) = P(A ∩ B)P(B) , provided P(B) ≠ 0 |
| 5 | Multiplication theorem (general) | P(A ∩ B) = P(A) · P(B A) = P(B) · P(A B) |
| 6 | Multiplication theorem (independent events) | P(A ∩ B) = P(A) · P(B) |
Independence is a stronger condition than mutual exclusiveness: two non-trivial events cannot be both. The NCERT example with a black king and a heart in a deck of 52 cards is the standard illustration the boards reuse.
Total Probability and Bayes' Theorem (The 5-Mark Long Answer)
The Probability Formula address this in the same order as the NCERT textbook.
The total probability theorem and Bayes' theorem together carry the heaviest mark allocation in the chapter. Every Class 12 student should be able to write both verbatim before substituting values from the stem.
| # | Concept | Formula |
|---|---|---|
| 7 | Total probability theorem | P(A) = i=1n P(Ei) · P(A Ei) , where E1, E2, , En partition the sample space |
| 8 | Bayes' theorem | P(Ei A) = P(Ei) · P(A Ei)j=1n P(Ej) · P(A Ej) |
Here E1, E2, are the prior partitions (factories, bags, machines), P(Ei) are the priors, P(A Ei) the likelihoods, and P(Ei A) the posterior. The denominator of Bayes is exactly the total-probability expression for event A.
Standard Bayes Procedure (write these five lines in order):
Step 1. Define E1, E2, (the priors) and the event A (the observation).
Step 2. List the priors P(Ei) from the stem.
Step 3. List the likelihoods P(A Ei) from the stem.
Step 4. Substitute into the Bayes formula above.
Step 5. Simplify the fraction and write the final probability.
Bernoulli Trials and the Binomial Distribution
The Probability Formula address this in the same order as the NCERT textbook.
The binomial distribution closes the chapter and is the second 5-mark target. A Bernoulli trial has exactly two outcomes (success or failure), trials are independent, and the success probability is constant across trials.
| # | Concept | Formula |
|---|---|---|
| 9 | Binomial probability of r successes in n trials | P(X = r) = nCr pr qn-r , where q = 1 - p and r = 0, 1, , n |
| 10 | Mean of the binomial distribution | μ = E(X) = np |
| 11 | Variance of the binomial distribution | σ2 = Var(X) = npq |
| 12 | Expected value of a discrete random variable | E(X) = i=1n xi · pi |
Variance npq is always less than the mean np , since q < 1 . If your answer says variance equals mean, you have mis-set the binomial. The expected value row applies to every discrete random variable and shows up in case-study questions on probability distributions.
How the Probability Formula on the Probability Formula Help You
This Collegedunia sheet condenses 30 pages of NCERT prose into a printable one-look reference, organised by the 5 concept blocks the CBSE paper actually tests.
- Concept-block layout matches the order CBSE markers expect: classical, conditional, multiplication, total-probability with Bayes, binomial distribution.
- Independence vs mutual exclusiveness is flagged in the conditional section, since this is the single most penalised conceptual error in board scripts.
- Bayes procedure is rendered as a five-step boxed sequence, matching the rubric examiners use for step marks.
- 2026-27 syllabus aligned with the current NCERT print; the dropped sub-topics are not included.
Probability Class 12 Important Formulae for Boards and JEE
The five rows below carry more than 75% of the board and JEE Main marks on this chapter. Drill these first.
| # | Formula | Where it is tested |
|---|---|---|
| 1 | P(A B) = P(A ∩ B)P(B) | Every Bayes step, every conditional case-study |
| 2 | P(A ∩ B) = P(A) · P(B A) | Multiplication rule, dependent trials |
| 3 | P(Ei A) = P(Ei) P(A Ei)j P(Ej) P(A Ej) | 5-mark Bayes long answer (factories, diseases, urns) |
| 4 | P(X = r) = nCr pr qn-r | 5-mark binomial long answer (defectives, tosses) |
| 5 | μ = np, σ2 = npq | 3-mark sub-part on mean and variance of a binomial |
Full reference: The complete 12-formula sheet, with symbol meanings and concept notes, is in the downloadable PDF above.
Common Mistakes Students Make in Probability Class 12
The Probability Formula are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
The errors below are flagged most often in CBSE marking-scheme reports for this chapter.
- Treating independence and mutual exclusiveness as the same condition. They are not.
- Writing only one term of the sum in the denominator of Bayes' theorem.
- Using q = p - 1 instead of q = 1 - p in the binomial formula.
- Swapping the prior probability and the likelihood. Re-read the stem before assigning P(Ei) and P(A Ei) .
- Skipping the explicit definition of the random variable X in a binomial answer. CBSE awards 1 step mark for the definition line alone.
Probability Notes and Solutions for Class 12 Maths Chapter 13
For solved board-style problems on each of the 12 formulas above, including the standard factory Bayes problem and the eight-toss binomial problem, use the companion resources below.
Also Check:
- NCERT Solutions for Class 12 Maths Chapter 13 Probability
- NCERT Notes for Class 12 Maths Chapter 13 Probability
- NCERT Exemplar Solutions for Class 12 Maths Chapter 13 Probability
NCERT Formula Sheet for Class 12 Maths: All Chapters
The full Class 12 Maths formula sheet library follows the same one-look layout, one PDF per chapter.
| Chapter | Formula Sheet |
|---|---|
| Chapter 1 | Relations and Functions Formula Sheet |
| Chapter 2 | Inverse Trigonometric Functions Formula Sheet |
| Chapter 3 | Matrices Formula Sheet |
| Chapter 4 | Determinants Formula Sheet |
| Chapter 5 | Continuity and Differentiability Formula Sheet |
| Chapter 6 | Application of Derivatives Formula Sheet |
| Chapter 7 | Integrals Formula Sheet |
| Chapter 8 | Application of Integrals Formula Sheet |
| Chapter 9 | Differential Equations Formula Sheet |
| Chapter 10 | Vector Algebra Formula Sheet |
| Chapter 11 | Three Dimensional Geometry Formula Sheet |
| Chapter 12 | Linear Programming Formula Sheet |
Probability Formula - Frequently Asked Questions
Probability Formula: 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 | Bayes' theorem and total probability |
| 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 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:
- 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 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.
- 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 definitions in this 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 these notes 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 marks this chapter contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
Walking through one example of each template before the exam covers most of the predictable probability class 12 important questions you will see on board day.
- these notes previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The this Class 12 page important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching the resource extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
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 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
The full the chapter notes 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 Notes Pair with NCERT Solutions and the Formula Sheet
The the PDF notes 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 Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| this chapter ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| probability class 12 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 these notes ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The probability class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The probability class 12 formulas reference sheet is the same A4 file students sometimes refer to as probability class 12 all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board this Class 12 page textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 probability solutions, class 12 probability ncert solutions, ncert class 12 probability solutions - 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 chapter cleanly:
| Reference | How it maps to the resource |
|---|---|
| RD Sharma Class 12 Probability | 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 the chapter notes textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Probability | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Probability Notes Page Most Effectively
The recommended study plan for the the PDF chapter 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 chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use the this chapter ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages 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 problem sets - sharpens speed and conceptual depth.
- The these notes important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
Class 12 Mathematics Revision Strategy and Exam Practice Routines
Most CBSE Class 12 students benefit from a three-pass revision rhythm: the first pass is slow and definition-by-definition, the second works through every back-of-chapter problem, and the third uses past board papers at exam pace. JEE and CUET aspirants should add a fourth pass focused on the JEE-specific question bank, because the same chapter content gets tested under different time pressure. Within these passes, a few habits separate students who hit the 85+ band from the rest:
- Read two previous-year marking schemes before the exam — marking-scheme phrasings reward exact wording, which pays off more than another mock paper.
- Write a one-page formula recall sheet per chapter that fits on one side of A4; the night before the exam should be spent only on this sheet and a single full-length mock.
- Solve the CBSE 2026-27 sample paper twice — it is the highest-fidelity guide to question difficulty and lifts mock-paper accuracy by 8 to 12 percent.
- Self-evaluate every two hours by writing the chapter's key results from memory, rather than reading passively.
- Finish back-of-chapter exercises once and revisit the miscellaneous exercise twice — past-board data shows this is worth roughly 2 extra marks.
Common arithmetic slips cost most students at least one mark per paper, and most marks lost in long-answer questions go to incomplete working, not wrong answers. Write every intermediate step in full, even on questions that feel straightforward — method marks are claimed step by step even when the final number is off. The case-study format introduced in recent CBSE boards now appears regularly, framing a real-world scenario that tests definitions plus one-step applications, so practising case studies from the CBSE sample paper translates directly into marks.
Time allocation in the last fortnight matters most. Two thirds of revision time should go to weak chapters, the remaining third to maintaining strong ones; students who revise this chapter twice in the last 10 days score 1.5 to 2 marks higher on past boards. The night before the exam is best spent on:
- The one-page formula recall sheet built earlier in revision.
- A single full-length mock paper at exam timing.
- Avoid learning any new material the night before — sleep matters more.
Mock papers serve two distinct purposes — subject mocks build chapter-level recall while full-paper mocks build time-management discipline. Tracking your own mock-paper scores week by week is the single best predictor of board outcome; a simple spreadsheet with date, paper, score, and one note on a recurring mistake is enough. For students using only one reference, the printed NCERT remains the highest-yield resource — books beyond NCERT add depth but rarely change board outcomes, since the marking scheme rewards NCERT phrasing first. Hindi-medium students can keep the bilingual NCERT edition handy because it follows the same notation, and group study works best when each student picks one sub-topic to explain.
Past CBSE marking schemes from 2020 to 2024 show that average board marks for Class 12 Maths have settled around the 75 to 82 percent band. Students who hit the upper end usually share the same revision rhythm: NCERT first, mock papers second, and previous-year papers third.
Frequently Asked Questions
Ques. What are all the formulas in these notes?
Ans. Chapter 13 holds 10 to 12 working formulas across 5 blocks: the classical definition and addition rule, conditional probability, the multiplication theorem, the total-probability theorem with Bayes' theorem, and the binomial distribution with its mean and variance. The full reference is in the downloadable PDF above.
Ques. What is the formula for Bayes' theorem in Class 12 NCERT?
Ans. Bayes' theorem states P(Ei A) = P(Ei) · P(A Ei)j P(Ej) · P(A Ej) , where the Ei partition the sample space. The denominator is exactly the total-probability expression for the event A.
Ques. What is the difference between independent and mutually exclusive events?
Ans. Independent events satisfy P(A ∩ B) = P(A) · P(B) and almost always have a non-empty intersection. Mutually exclusive events satisfy P(A ∩ B) = 0 and so cannot both occur. Two non-trivial events cannot be both independent and mutually exclusive.
Ques. What is the binomial distribution formula in Class 12 Probability?
Ans. For n Bernoulli trials with success probability p, the probability of exactly r successes is P(X = r) = nCr pr qn-r , where q = 1 - p . The mean is np and the variance is npq .
Ques. How many marks does Probability carry in CBSE Class 12 Maths?
Ans. Probability typically carries 8 to 10 marks in the Class 12 Maths board paper, distributed across a 5-mark Bayes or binomial long answer, a 3-mark conditional-probability case-study, and a short MCQ or fill-in.
Ques. What is the formula for conditional probability?
Ans. The conditional probability of A given B is P(A B) = P(A ∩ B)P(B) , provided P(B) ≠ 0 . The denominator is the conditioning event, never a multiplier.
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