Probability Class 12 Notes - Class 12 Maths Ch 13

This page hosts the this chapter, presenting it for Class 12 Mathematics Chapter 13 Probability. These notes are sectioned to match the NCERT chapter, with each section closing in a short summary box and a single solved example. Suitable for both board and JEE Main preparation, the notes are free to download.

• CBSE Boards 2025: 8 to 10 marks, with one 5-mark Bayes question and one 2-mark conditional-probability item appearing in almost every set.
• JEE Main 2025: 2 to 3 questions per shift across Bayes, independence, and the Binomial distribution. Bayes is the single highest-yield sub-topic for JEE Main.

These Collegedunia this Class 12 page are cross-verified against the latest NCERT print, the CBSE 2025 marking scheme, and the JEE Main 2024 and 2025 answer keys. The numbered solved example inside every section mirrors the question style that examiners reward in board and JEE Main papers.

Also Check:

• Probability Class 12 Maths NCERT Solutions
• Probability Class 12 Maths Formula Sheet

Probability Class 12 Notes: What the 2026-27 NCERT Chapter Covers

Chapter 13 in the 2026-27 NCERT Class 12 Mathematics textbook is built around five teachable blocks. The first defines conditional probability and proves its three basic properties. The second derives the multiplication theorem and the independence test.

The third sets up the total probability theorem and uses it to prove Bayes' theorem. The fourth introduces a random variable, its probability distribution, mean, and variance. The fifth covers Bernoulli trials and the Binomial distribution, including its mean np and variance npq .

The book lists 5 solved examples in the conditional-probability section, 7 in the Bayes block, and 6 in the Binomial section, alongside 4 graded exercises (13.1, 13.2, 13.3 miscellaneous) and 1 summary box. These Class 12 probability notes follow the same section order so the revision matches the the resource page-by-page.

The 9 Formulas You Must Reproduce From Memory

Every JEE Main and CBSE Boards question in Probability collapses into one of these 9 expressions. If a student can write and apply them under timed conditions, the chapter notes is secured.

Full version: Probability Class 12 Formula Sheet with derivations and 14 additional identities.

Probability Video Walkthrough

Source: Magnet Brains on YouTube

Conditional Probability and the Multiplication Theorem

Conditional probability measures the chance of event A given that event B has already occurred. The defining identity P(A|B) = P(A ∩ B)/P(B) only applies when P(B) > 0 .

Three textbook properties follow: P(S|B) = 1 , P(A' |B) = 1 - P(A|B) , and P((A ∪ C)|B) = P(A|B) + P(C|B) - P((A ∩ C)|B) .

The multiplication theorem rearranges the definition into P(A ∩ B) = P(A) P(B|A) . For three events: P(A ∩ B ∩ C) = P(A) P(B|A) P(C|A ∩ B) . Two events are independent when P(A ∩ B) = P(A) P(B) , which is equivalent to P(A|B) = P(A) . Mutually exclusive events with non-zero probability can never be independent.

Solved example. A bag has 3 red and 5 black balls. Two are drawn without replacement. P(both red) = 38 · 27 = 328 . The second factor is conditional because removing the first red ball changes the sample space.

Total Probability Theorem and Bayes' Theorem

Let A₁, A₂, , A_n be a partition of the sample space (mutually exclusive, exhaustive, each with non-zero probability). For any event E , the total probability theorem gives P(E) = _i=1ⁿ P(A_i) P(E|A_i) . Bayes' theorem inverts the conditioning to compute the posterior P(A_i|E) once E is observed.

Bayes is the single most-asked sub-topic of the the PDF. CBSE has set a 5-mark Bayes question in 9 of the past 10 board papers, and JEE Main 2025 carried one Bayes MCQ in 4 of 6 shifts. The standard template: identify the partition (often urns, machines, or test outcomes), write each prior P(A_i) , the likelihoods P(E|A_i) , and then divide.

Solved problem. Two factories supply bulbs in the ratio 60:40. The first has a 2% defect rate, the second 5%. A bulb tested is defective. The probability it came from the second factory is P(F₂|D) = 0.40 × 0.050.60 × 0.02 + 0.40 × 0.05 = 0.0200.032 = 0.625.

Random Variables, Probability Distribution, Mean and Variance

A random variable assigns a real number to every outcome in the sample space. Its probability distribution lists each value x_i with probability p_i , subject to ∑ p_i = 1 .

The mean (expectation) is E(X) = ∑ x_i p_i . The variance is Var(X) = E(X²) - [E(X)]² , and the standard deviation is its positive square root.

A common JEE Main slip: students compute ∑ (x_i - μ)² instead of ∑ (x_i - μ)² p_i . The probabilities act as weights and must multiply each squared deviation. Tabulating x_i, p_i, x_i p_i, x_i² p_i in four columns avoids this every time.

Bernoulli Trials and the Binomial Distribution

A Bernoulli trial has exactly two outcomes (success, failure), a fixed success probability p , and is independent of every other trial.

A Binomial distribution B(n, p) counts the number of successes in n such trials. Its PMF is P(X = r) = ⁿC_r p^r q^n-r where q = 1 - p . Mean = np , variance = npq , and npq ≤ np always.

Three trial-conditions check before declaring a problem Binomial: trials are independent, n is fixed in advance, and p does not change across trials. Sampling without replacement breaks the third condition and is not Binomial.

5 Most-Repeated Question Patterns in CBSE Boards and JEE Main

The same five templates carry over 80% of the marks in the this chapter across CBSE 2021 to 2025 and JEE Main 2023 to 2025. Recognising the template is half the work; the formula plug-in is the other half.

Previous Year Trend: Probability in CBSE and JEE Main

The table tracks the appearance and mark-weight of each sub-topic across the past five years of CBSE Class 12 Maths and JEE Main. Bayes' theorem and the Binomial distribution share the top spot; conditional probability never drops below 2 marks in any board paper.

Full year-wise PYQ map: the chapter notes NCERT Solutions.

How Will Collegedunia's Probability Notes Help You Score Higher?

These Collegedunia notes are calibrated to the way CBSE and JEE Main actually frame Probability questions. The Bayes section walks through the urn-factory-test template that has appeared in 9 of the past 10 board papers.

The Binomial section ends with the three trial-condition checks that catch the 90% of students who misapply the formula to a sampling-without-replacement setup. Every formula sits beside the question pattern that uses it, so reading these notes once is equivalent to drilling 20 mixed problems.

Common Mistakes Students Make When Using the Probability Class 12 Notes

• Treating mutually exclusive events as independent. They are independent only when at least one has zero probability.
• Forgetting the weights p_i in the variance formula and computing ∑ (x_i - μ)² instead.
• Applying the Binomial PMF to sampling without replacement.
• Confusing the prior P(A_i) with the posterior P(A_i | E) in a Bayes setup.
• Dropping the conditioning event in P(A | B) + P(A' | B) = 1 and writing P(A) + P(A') instead.

Related Resources for Class 12 Maths Chapter 13 Probability

• this Class 12 page NCERT Solutions
• the resource Formula Sheet
• the chapter notes Handwritten Notes
• the PDF Exemplar Solutions

NCERT Notes for Class 12 Maths: All Chapters

The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.

this Class 12 page: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.

Probability Class 12 Notes - Quick Summary

• The the resource cover every section of Class 12 Mathematics Chapter 13 Probability, aligned to the 2026-27 NCERT print.
• The chapter notes include formal definitions, solved examples and end-of-section formula recap suitable for board and JEE Main preparation.
• The the PDF are downloadable as a free PDF and follow the notation of the official NCERT textbook line for line.

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.

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:

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:

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.

• the PDF 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 solutions set is reused by toppers in the last fortnight of revision.
• For NCERT Exemplar practice, the matching this chapter 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:

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 Notes Pair with NCERT Solutions and the Formula Sheet

The Probability Class 12 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:

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 this Class 12 page 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 the resource 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:

How to Use the Probability Notes Page Most Effectively

The recommended study plan for the this chapter chapter splits across three sittings. The table below outlines what to do in each:

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 probability class 12 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.