NCERT Solutions for Class 12 Maths Chapter 13 Probability
Probability in Class 12 is noticeably different from what students met in earlier classes, because the chapter now deals with conditional probability, dependent events, and the idea that an earlier outcome can change the likelihood of a later one. Myclass24 has prepared these NCERT Solutions for Class 12 Maths Chapter 13 to make that shift in thinking clear from the very first exercise, rather than letting students carry forward Class 10-style intuition into problems where it no longer applies. Each solution explains the reasoning behind choosing a formula, not just the formula itself.
Summary of Chapter 13 Probability
Basics of probability, Problem on P & C. Introduction, Conditional Probability, multiplication theorem on probability, independent events, Total probability: Examples, Bayes’ theorem independent events, Total probability: Examples, Bayes’ theorem, the Probability distribution of a random variable, Mean & Variance of distribution, Binomial Distribution
Find the PDF of NCERT Solutions for Class 12 Maths Chapter 13
Probability questions in board exams frequently combine multiple concepts: conditional probability followed by Bayes' theorem, or independence checks followed by a binomial distribution question, which makes having a single organised reference genuinely helpful while revising. The Myclass24 PDF for this chapter sets out each exercise with the formula used clearly stated before the working begins, so during a quick revision pass you can match a question type to the right approach without re-deriving everything from scratch.
Class | 12 |
Subject | Mathematics |
Chapter Number | 13 |
Chapter Name | Probability |
Board | CBSE / NCERT |
Total Exercises | 5 (including Miscellaneous Exercise) |
Key Topics | Conditional probability, multiplication theorem, independent events, Bayes' theorem, random variables and probability distributions, mean and variance of a random variable, Bernoulli trials, binomial distribution |
Weightage | Around 8 to 10 marks, one of the higher-weightage chapters in the board exam |
Difficulty Level | Moderate to high, especially Bayes' theorem and binomial distribution |
Prerequisite Chapter | Class 11 Probability and basic set theory |
Understanding Chapter 13: The Concepts That Decide Your Score
Conditional probability is the foundation of this entire chapter, and the single most common error students make is misreading which event is given and which event is being asked about. The notation P(A given B) is not the same as P(B given A), and mixing these up changes the entire answer even though the arithmetic afterwards looks identical. Myclass24's solutions consistently restate, in words, what each conditional probability expression means in the context of the actual question before any calculation begins, because that one sentence of clarity prevents the most common mistake in the chapter.
Bayes' theorem is where many students lose confidence, mainly because the formula looks intimidating with its multiple terms in the denominator. Our solutions break every Bayes' theorem question into the same three stages every time: first list the prior probabilities of each cause, then list the conditional probabilities of the observed event given each cause, and only then assemble the formula. Once a student sees this three-stage pattern repeated across different questions- lottery tickets, factory defects, disease testing- the formula stops being intimidating and becomes a routine three-step process.
The second half of the NCERT Solutions for Class 12 chapter, random variables and the binomial distribution, asks students to shift from finding a single probability to building an entire probability distribution table. A frequent slip is forgetting that the probabilities in this table must sum to exactly 1, which is actually a useful self-check students rarely use to verify their own working. Myclass24's solutions include this check explicitly wherever a probability distribution table is constructed, turning a verification step into a habit rather than an afterthought.
For Bernoulli trials and the binomial distribution specifically, the chapter only works cleanly when three conditions hold: a fixed number of trials, only two possible outcomes per trial, and a constant probability of success across all trials. Our solutions point out, wherever relevant, why a given scenario does or does not qualify as a binomial setting, because CBSE has increasingly favoured questions that test this conceptual recognition rather than pure formula substitution. Spotting the binomial structure correctly is, in the end, more valuable than memorising the formula for P(X = r).
NCERT Solutions for Class 12 Maths Chapter 13 – Probability FAQs
Conditional probability is the probability of an event occurring when it is known that another event has already occurred. It helps students understand how additional information can affect the likelihood of an outcome. In Chapter 13, conditional probability is one of the most important concepts and forms the basis for several advanced topics. NCERT exercises include a variety of questions involving real-life situations, events, and probability calculations. Understanding conditional probability improves logical reasoning and analytical thinking. A clear grasp of this concept enables students to solve complex probability problems accurately and lays a strong foundation for higher studies in mathematics and statistics.
Bayes’ Theorem is a fundamental result in probability that helps determine the probability of an event based on prior information. It is particularly useful when multiple possible causes can lead to the same outcome. In this chapter, students learn how to apply the theorem using conditional probabilities. NCERT exercises often present practical situations involving selection, testing, and classification problems. Understanding Bayes’ Theorem helps students analyse probability scenarios systematically and reach accurate conclusions. Regular practice of theorem-based questions improves confidence and problem-solving skills, making it easier to tackle both board examination questions and competitive-level probability problems.
The multiplication theorem provides a method for finding the probability of two events occurring together. It establishes a relationship between joint probability and conditional probability. Students learn how to apply the theorem to both independent and dependent events. NCERT exercises contain several examples where the multiplication rule simplifies calculations and helps solve multi-step probability questions. Understanding this theorem is essential because it serves as a building block for many advanced concepts discussed in the chapter. By practising a variety of problems, students develop a stronger understanding of event relationships and improve their ability to solve probability-based questions efficiently.
Determining whether events are independent or dependent is a common requirement in probability questions. Independent events are those where the occurrence of one event does not affect the probability of the other. Dependent events, on the other hand, influence each other's outcomes. NCERT exercises include numerous situations where students must analyse event relationships before applying formulas. Understanding the distinction is important because the method of calculation changes depending on the type of event. Careful reading of the question and proper interpretation of the conditions help students choose the correct approach and avoid common calculation errors.
Many students make mistakes by applying incorrect probability formulas or failing to identify whether events are independent or dependent. Another common error is misunderstanding the information provided in the question and selecting the wrong event definitions. Some students also make calculation mistakes while simplifying fractions and probability expressions. NCERT exercises encourage students to approach problems systematically by defining events clearly before performing calculations. Drawing diagrams, tables, or probability trees can also improve understanding. Consistent practice and careful interpretation of questions help students reduce errors, strengthen conceptual clarity, and perform well in board examinations and competitive mathematics tests.