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CBSE Class 12 Mathematics Chapter 13 Probability Worksheet with Answers

Probability is a pivotal chapter in CBSE Class 12 Mathematics, contributing 10 marks to the board exam and forming the foundation for higher studies in statistics, data science, and engineering. This printable worksheet is meticulously designed to align with the latest NCERT syllabus and CBSE exam pattern for 2025. It features a blend of objective and subjective questions that test your grasp of conditional probability, multiplication theorem, Bayes' theorem, and random variables. Complete this worksheet in 90 minutes to sharpen your exam temperament and identify areas needing revision.

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Key takeaways

✓Chapter 13 Probability carries 10 marks in the CBSE Class 12 board exam and covers conditional probability, multiplication theorem, Bayes' theorem, and random variables.
✓This worksheet contains 30+ graded questions across six sections—MCQs, fill-in-the-blanks, match columns, short answers, long HOTS problems, and one case-study question.
✓Difficulty level is Moderate to Challenging; allocate 90 minutes for completion to simulate board exam conditions.
✓Every question is accompanied by a detailed solution in the answer key section, enabling effective self-evaluation and concept reinforcement.
✓Conditional probability formula P(A|B) = P(A ∩ B) / P(B) and Bayes' theorem are frequently tested in both objective and subjective formats.
✓Random variables, probability distributions, mean and variance calculations form the backbone of long-answer questions worth 4–6 marks each.
✓Regular practice with such worksheets significantly improves speed, accuracy, and confidence for the actual CBSE board examination.

Quick Chapter Recap: Probability for Class 12

Chapter 13 Probability builds on the foundational concepts introduced in Class 11 and introduces advanced topics essential for board exams and competitive tests like JEE. Conditional probability is the probability of an event A given that event B has already occurred, expressed as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. The multiplication theorem states P(A ∩ B) = P(A) × P(B|A) or P(B) × P(A|B), a critical tool for solving dependent event problems. Bayes' theorem allows you to reverse conditional probabilities and is widely used in diagnostics, machine learning, and decision theory. Its formula is P(E_i | A) = [P(E_i) × P(A | E_i)] / Σ[P(E_j) × P(A | E_j)]. Random variables assign numerical values to outcomes of a random experiment; they can be discrete or continuous. For a discrete random variable X, the probability distribution satisfies Σ P(X = x_i) = 1. The mean (expectation) is μ = Σ x_i P(X = x_i) and variance is σ² = Σ (x_i − μ)² P(X = x_i) or Σ x_i² P(X = x_i) − μ². These formulas appear regularly in 4-mark and 6-mark questions, so memorize them thoroughly.

Conditional probability: P(A|B) = P(A ∩ B) / P(B) when P(B) ≠ 0
Multiplication theorem: P(A ∩ B) = P(A) P(B|A) = P(B) P(A|B)
Bayes' theorem: enables calculation of reverse conditional probabilities using partition theorem
Random variable: a function mapping sample space outcomes to real numbers
Mean of X: E(X) = Σ x_i P(X = x_i); Variance: Var(X) = E(X²) − [E(X)]²
Total probability theorem: P(A) = Σ P(E_i) P(A | E_i) for mutually exclusive and exhaustive events E_i

Worksheet Instructions and Difficulty Level

This worksheet is structured to mirror the actual CBSE Class 12 board exam pattern for Chapter 13 Probability. It contains six distinct sections that progressively test recall, application, analysis, and higher-order thinking skills. Section A comprises 6 multiple-choice questions (1 mark each); Section B has 5 fill-in-the-blank items (1 mark each); Section C offers match-the-following or true/false (1 mark each); Section D contains 5 short-answer questions (2 marks each); Section E features 3 long-answer or HOTS questions (4–6 marks each); and one case-study question (4 marks with sub-parts). The difficulty level is Moderate to Challenging, designed for students aiming to score 90+ in Mathematics. Suggested time is 90 minutes under exam conditions—no calculator, no notes. Attempt all questions in sequence, marking any doubts for later review. Use the detailed answer key at the end to verify your solutions and understand step-by-step reasoning. Regular practice with such worksheets, alongside NCERT Class 12 Mathematics textbook exercises and exemplar problems, will consolidate your understanding and boost exam confidence.

Total marks: 40 | Total questions: 30+ across six sections
Difficulty: Moderate to Challenging, aligned with 2025 CBSE board exam pattern
Time allocation: 90 minutes (strictly timed for exam simulation)
Section A (MCQs): 6 questions × 1 mark = 6 marks
Section B (Fill-in-the-blanks): 5 questions × 1 mark = 5 marks
Section C (Match/True-False): 1 question × 5 marks = 5 marks
Section D (Short answer): 5 questions × 2 marks = 10 marks
Section E (Long answer/HOTS): 3 questions × 4–6 marks = 14 marks
Case study: 1 question with 4 sub-parts = 4 marks

Section A: Multiple Choice Questions (1 mark each)

1. If P(A) = 0.4, P(B) = 0.5, and P(A ∪ B) = 0.7, then P(A ∩ B) equals: (a) 0.1 (b) 0.2 (c) 0.3 (d) 0.4 2. For two independent events A and B, if P(A) = 0.3 and P(B) = 0.6, then P(A ∩ B) is: (a) 0.18 (b) 0.9 (c) 0.3 (d) 0.6 3. A bag contains 5 red and 3 blue balls. Two balls are drawn at random without replacement. The probability that both are red is: (a) 5/14 (b) 10/28 (c) 5/16 (d) 25/64 4. If P(A|B) = 0.8, P(B) = 0.5, then P(A ∩ B) equals: (a) 0.3 (b) 0.4 (c) 1.3 (d) 0.16 5. The probability distribution of a random variable X is given by P(X = x) = kx for x = 1, 2, 3, 4. The value of k is: (a) 1/10 (b) 1/8 (c) 1/5 (d) 1/6 6. Two dice are thrown. Let A be the event 'sum is 9' and B be 'first die shows 5'. Then P(A|B) is: (a) 1/6 (b) 1/9 (c) 4/36 (d) 1/3

Section B: Fill in the Blanks (1 mark each)

7. If events A and B are mutually exclusive, then P(A ∩ B) = __________. 8. The formula for conditional probability P(A|B) is __________ / P(B). 9. For independent events E and F, P(E ∩ F) = P(E) × __________. 10. If P(A) = 0.6, P(B) = 0.4, and P(A ∩ B) = 0.24, then events A and B are __________ (independent/dependent). 11. The sum of all probabilities in a probability distribution of a random variable X equals __________. These fill-in-the-blank questions test your recall of fundamental definitions and properties from NCERT Class 12 Mathematics Chapter 13. They are quick checkpoints to ensure you have internalized core terminology before moving to application-based problems. Each blank expects a precise mathematical expression or keyword, so revise your Class 12 Mathematics notes carefully.

Section C: Match the Following (5 marks)

12. Match Column I with Column II: Column I (Concept) A. P(A|B) B. P(A ∩ B) for independent events C. Bayes' theorem application D. Variance of X E. Total probability theorem Column II (Formula / Description) 1. P(A) × P(B) 2. E(X²) − [E(X)]² 3. P(A ∩ B) / P(B) 4. P(E_i | A) using prior probabilities 5. P(A) = Σ P(E_i) P(A | E_i) Write your answers as A–__, B–__, C–__, D–__, E–__. This section assesses your ability to connect definitions with their mathematical expressions and applications. Matching exercises like this frequently appear in CBSE 12 Mathematics board papers and help reinforce conceptual linkages. Ensure you have thoroughly read the NCERT Class 12 Mathematics textbook section on conditional probability, Bayes' theorem, and random variables before attempting this question. Each correct match earns 1 mark.

Section D: Short Answer Questions (2 marks each)

13. A coin is tossed three times. Let A be the event 'at least two heads' and B be 'first toss is head'. Find P(A|B). 14. Two cards are drawn successively without replacement from a well-shuffled deck of 52 cards. Find the probability that both are kings. 15. If P(A) = 0.5, P(B) = 0.4, and P(A ∪ B) = 0.7, check whether A and B are independent. 16. A random variable X has the probability distribution: X: 0 1 2 P(X): 0.3 0.5 0.2 Find the mean of X. 17. In a factory, machine M1 produces 30% and machine M2 produces 70% of the total output. The defective percentages are 5% for M1 and 2% for M2. If a randomly selected item is defective, find the probability it came from M1 (use Bayes' theorem). These short-answer questions carry 2 marks each and require clear step-by-step working. Show all formulae used, substitute values correctly, and box your final answer. CBSE Class 12 board examiners award partial marks for method, so never leave a question blank.

Section E: Long Answer and HOTS Questions (4–6 marks each)

18. (6 marks) A bag contains 4 red, 5 black, and 6 white balls. Three balls are drawn at random without replacement. Find the probability that: (i) all three are of the same colour (ii) one is of each colour (iii) at least one is red. 19. (4 marks) A random variable X has the following probability distribution: X: −2 −1 0 1 2 P(X): 0.1 k 0.2 2k 0.3 Find: (i) the value of k (ii) mean of X (iii) variance of X. 20. (4 marks) In a certain test, 60% of students pass. A student who studies has a 90% chance of passing, while a student who does not study has a 30% chance of passing. A randomly selected student passes. What is the probability that the student had studied? (Apply Bayes' theorem.) Long-answer questions demand logical structuring, correct application of theorems, and numerical accuracy. CBSE examiners look for clarity in presentation—use proper notation, label all intermediate steps, and write conclusions explicitly. These HOTS (Higher Order Thinking Skills) problems often combine concepts from conditional probability, Bayes' theorem, and random variables, mirroring the toughest questions in the actual board paper. Practice such multi-step problems regularly using NCERT Class 12 Mathematics exemplar and previous years' board papers.

Section F: Case-Study Based Question (4 marks)

21. Read the following case study and answer the sub-questions: A pharmaceutical company tests a new drug for a disease that affects 2% of the population. The test correctly identifies a diseased person 95% of the time (sensitivity) and correctly identifies a healthy person 90% of the time (specificity). (i) (1 mark) What is the probability that a randomly selected person from the population has the disease? (ii) (1 mark) If a person tests positive, write the expression for P(Disease | Positive) using Bayes' theorem (do not solve). (iii) (2 marks) Calculate the probability that a person who tests positive actually has the disease. Case-study questions are a staple of the CBSE Class 12 Mathematics board exam from 2021 onwards. They assess your ability to extract mathematical information from real-world scenarios and apply Probability concepts like Bayes' theorem, conditional probability, and the total probability theorem. Read the case carefully, identify given probabilities, define events clearly (e.g., D for disease, P for positive test), and then substitute into the standard formulae. Show all working to earn full marks. These questions typically carry 4–5 marks and are considered scoring if you practice enough case-studies from NCERT exemplar and sample papers.

Complete Answer Key with Explanations

Section A Answers (MCQs): 1. (b) 0.2. P(A ∩ B) = P(A) + P(B) − P(A ∪ B) = 0.4 + 0.5 − 0.7 = 0.2. 2. (a) 0.18. For independent events, P(A ∩ B) = P(A) × P(B) = 0.3 × 0.6 = 0.18. 3. (a) 5/14. P(both red) = (5/8) × (4/7) = 20/56 = 5/14. 4. (b) 0.4. P(A ∩ B) = P(A|B) × P(B) = 0.8 × 0.5 = 0.4. 5. (a) 1/10. Σ P(X = x) = k(1+2+3+4) = 10k = 1 ⇒ k = 1/10. 6. (a) 1/6. P(A ∩ B) = P{(5,4)} = 1/36; P(B) = 6/36 = 1/6; P(A|B) = (1/36)/(1/6) = 1/6. Section B Answers (Fill-in-the-blanks): 7. 0 (mutually exclusive events have no common outcomes). 8. P(A ∩ B) (definition of conditional probability). 9. P(F) (independence implies P(E ∩ F) = P(E) P(F)). 10. independent (since 0.6 × 0.4 = 0.24 = P(A ∩ B)). 11. 1 (axiom of probability distribution). Section C Answer (Match the Following): 12. A–3, B–1, C–4, D–2, E–5. Section D Answers (Short answer): 13. Sample space for B = {HHH, HHT, HTH, HTT}; |B| = 4. A ∩ B = {HHH, HHT, HTH}; |A ∩ B| = 3. P(A|B) = 3/4. 14. P(both kings) = (4/52) × (3/51) = 12/2652 = 1/221. 15. P(A) P(B) = 0.5 × 0.4 = 0.2. P(A ∩ B) = P(A) + P(B) − P(A ∪ B) = 0.5 + 0.4 − 0.7 = 0.2. Since they are equal, A and B are independent. 16. E(X) = 0×0.3 + 1×0.5 + 2×0.2 = 0 + 0.5 + 0.4 = 0.9. 17. Let D = defective. P(M1) = 0.3, P(M2) = 0.7, P(D|M1) = 0.05, P(D|M2) = 0.02. P(D) = 0.3×0.05 + 0.7×0.02 = 0.015 + 0.014 = 0.029. P(M1|D) = (0.3 × 0.05)/0.029 = 0.015/0.029 ≈ 0.517. Section E Answers (Long answer): 18. Total balls = 15. (i) P(all same) = [C(4,3) + C(5,3) + C(6,3)] / C(15,3) = [4+10+20]/455 = 34/455. (ii) P(one each) = (4×5×6)/C(15,3) = 120/455 = 24/91. (iii) P(at least 1 red) = 1 − P(no red) = 1 − C(11,3)/C(15,3) = 1 − 165/455 = 290/455 = 58/91. 19. (i) 0.1 + k + 0.2 + 2k + 0.3 = 1 ⇒ 3k = 0.4 ⇒ k = 0.4/3 ≈ 0.133. (ii) E(X) = (−2×0.1) + (−1×k) + 0×0.2 + 1×2k + 2×0.3 = −0.2 − k + 2k + 0.6 = 0.4 + k = 0.4 + 0.133 ≈ 0.533. (iii) E(X²) = 4×0.1 + 1×k + 0 + 1×2k + 4×0.3 = 0.4 + 3k + 1.2 = 1.6 + 3×0.133 ≈ 2.0. Var(X) = E(X²) − [E(X)]² ≈ 2.0 − (0.533)² ≈ 1.716. 20. Let S = studied, P = pass. P(S) unknown, assume 50% study, so P(S) = 0.5, P(S') = 0.5. P(P|S) = 0.9, P(P|S') = 0.3. P(P) = 0.5×0.9 + 0.5×0.3 = 0.45 + 0.15 = 0.6. P(S|P) = (0.5×0.9)/0.6 = 0.45/0.6 = 0.75. Section F Answer (Case study): 21. (i) P(Disease) = 0.02. (ii) P(Disease|Positive) = [P(Disease) × P(Positive|Disease)] / [P(Disease)×P(Positive|Disease) + P(Healthy)×P(Positive|Healthy)]. (iii) P(Positive|Disease) = 0.95, P(Healthy) = 0.98, P(Positive|Healthy) = 0.10. P(Disease|Positive) = (0.02×0.95)/(0.02×0.95 + 0.98×0.10) = 0.019/(0.019+0.098) = 0.019/0.117 ≈ 0.162 or 16.2%.

Every answer includes the formula or theorem applied, making self-study effective
Partial marks are awarded in CBSE board exams for correct methodology, so show all steps
Cross-check numerical answers using a calculator after completion
For Bayes' theorem problems, always write the total probability in the denominator explicitly
Variance calculation: remember Var(X) = E(X²) − [E(X)]² to avoid sign errors
If an MCQ stumps you, eliminate obviously wrong options and guess smartly to save time

How to Use This Worksheet for Maximum Benefit

Print this worksheet on A4 sheets and attempt it in one sitting under strict exam conditions—no mobile, no notes, 90-minute timer. Use a pencil for rough work on the margins and write final answers in pen to simulate board exam discipline. After completing all sections, compare your solutions with the detailed answer key provided. Award yourself marks honestly, deducting half-marks for calculation errors or missing steps even if the final answer is correct, because CBSE marking schemes penalize incomplete reasoning. Identify recurring mistakes—common pitfalls include misapplying Bayes' theorem, forgetting to verify Σ P(X = x_i) = 1 in random variable questions, or confusing P(A|B) with P(B|A). Maintain an error log in your Class 12 Mathematics notes and revise those concepts using the NCERT textbook or NCERT exemplar. For persistent doubts, upload a photo of the problem to CBSETUTOR.ai, where a 24×7 AI tutor provides step-by-step solutions instantly at just ₹999/month for all subjects and classes (6–12)—one flat price, 3-day free trial. Repeat this worksheet after one week to track improvement. Pair it with at least three other Probability worksheets from different sources, CBSE sample papers, and previous years' board questions to cover all question patterns. Regular timed practice builds speed, reduces silly mistakes, and ensures you finish the 10-mark Probability section confidently in the actual board exam.

Attempt the worksheet in one 90-minute sitting to build exam stamina and time management
Self-assess using the detailed answer key; deduct marks for incomplete steps, not just wrong answers
Maintain an error log for recurring mistakes—review NCERT Class 12 Mathematics textbook for clarification
Upload difficult problems to CBSETUTOR.ai for instant, step-by-step AI tutor support at ₹999/month (3-day free trial)
Redo this worksheet after one week to measure retention and concept mastery
Combine with CBSE sample papers, previous years' questions, and NCERT exemplar for comprehensive preparation
Focus on high-weightage topics: Bayes' theorem (4–6 marks), random variables and distributions (4–6 marks)

Why Chapter 13 Probability Matters for CBSE Class 12 Boards and Beyond

Probability consistently contributes 10 marks to the CBSE Class 12 Mathematics board paper, typically distributed as one 2-mark question, one 4-mark question, and one 6-mark question or case-study. Mastery of this chapter is non-negotiable for students targeting 90+ overall, because Probability questions are often straightforward if you know the formulae and practice diverse problem types. Beyond boards, Probability forms the backbone of undergraduate courses in statistics, economics, engineering, computer science, and data science. Concepts like conditional probability and Bayes' theorem are directly applied in machine learning algorithms, medical diagnostics, risk assessment, and artificial intelligence. For JEE Main and Advanced aspirants, Probability questions test logical reasoning and combinatorial thinking under time pressure. NEET does not have Mathematics, but students appearing for CUET, state engineering entrance exams, and actuarial science certifications will find this chapter indispensable. The 2025 CBSE Class 12 board exam will likely feature one case-study question on Probability worth 4–5 marks, so practicing real-world scenarios—like disease testing, quality control, or game theory—is crucial. This worksheet mirrors that exam pattern, ensuring you are well-prepared for any question variant. Dedicate at least two weeks exclusively to Probability revision, solving NCERT exercises, exemplar problems, and multiple worksheets like this one to cement your conceptual clarity and computational speed.

Probability carries a fixed 10 marks in the CBSE Class 12 Mathematics board exam every year
Typical breakup: one 2-mark, one 4-mark, and one 6-mark or case-study question
High-scoring chapter if formulas are memorized and diverse problem types practiced regularly
Essential foundation for undergraduate statistics, data science, machine learning, and actuarial science
JEE Main and Advanced regularly feature 1–2 Probability questions combining logic and combinatorics
CUET, state entrance exams, and BITSAT also test Chapter 13 concepts extensively
Case-study questions (introduced in 2021 CBSE pattern) demand real-world application of Bayes' theorem and conditional probability

Frequently asked questions

How many marks does Chapter 13 Probability carry in the CBSE Class 12 board exam?+

Probability contributes exactly 10 marks to the CBSE Class 12 Mathematics board paper, usually split across one 2-mark, one 4-mark, and one 6-mark question or a case-study question worth 4–5 marks.

What is the difference between P(A|B) and P(B|A)?+

P(A|B) is the probability of event A occurring given that B has occurred, calculated as P(A ∩ B)/P(B). P(B|A) is the reverse: probability of B given A, equal to P(A ∩ B)/P(A). They are generally not equal unless A and B are independent.

When should I use Bayes' theorem in a Probability question?+

Use Bayes' theorem when you need to find P(E_i | A)—the probability of a cause given an observed effect. It requires prior probabilities P(E_i) and likelihoods P(A | E_i), and is common in medical tests, quality control, and diagnostic problems.

How do I verify if two events are independent?+

Events A and B are independent if and only if P(A ∩ B) = P(A) × P(B). Alternatively, check if P(A|B) = P(A) or P(B|A) = P(B). If any equality holds, they are independent; otherwise, they are dependent.

What is the quickest way to calculate variance of a random variable X?+

Use the formula Var(X) = E(X²) − [E(X)]². First compute E(X) = Σ x_i P(X = x_i), then E(X²) = Σ x_i² P(X = x_i), and subtract the square of E(X). This method reduces calculation errors compared to the definition Σ (x_i − μ)² P(X = x_i).

Are case-study questions in Probability difficult?+

Case-study questions test the same concepts—conditional probability, Bayes' theorem, random variables—but wrapped in a real-world scenario. Read carefully, define events clearly, extract given probabilities, and apply standard formulas. With practice, they become highly scoring.

How much time should I spend on the Probability section during the board exam?+

Allocate roughly 25–30 minutes for the 10-mark Probability section in the 3-hour paper. Solve the shorter 2-mark question first to build confidence, then tackle the longer problems. Practice timed worksheets to improve speed and accuracy.

Can I score full marks in Probability if I only practice NCERT exercises?+

NCERT exercises cover fundamental concepts, but CBSE board exams include case-studies, HOTS questions, and multi-step problems. Supplement NCERT with exemplar problems, sample papers, previous years' questions, and worksheets like this to ensure comprehensive preparation.

What are the most common mistakes students make in Probability?+

Common errors include confusing P(A|B) with P(B|A), forgetting to check Σ P(X = x_i) = 1, misapplying Bayes' theorem by omitting the total probability denominator, and calculation slips in variance. Maintain an error log and revise regularly.

Where can I get instant help if I am stuck on a Probability problem?+

Upload a photo of the problem to CBSETUTOR.ai, where a 24×7 AI tutor provides step-by-step solutions for all CBSE Class 12 Mathematics chapters. It costs just ₹999/month for unlimited access to all subjects (Classes 6–12) with a 3-day free trial.

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