CBSE Class 12 Probability PYQs Analysis (2021-2025)

Based on detailed analysis of last 5 years' papers. Perfect for 2026 Boards prep!

Key Trends

Weightage Breakdown: 1-2 MCQs/assertion-reason (1 mark), 1-2 short answers (2-3 marks), 1-2 long answers on Bayes'/binomial/RV (4-6 marks), occasional case-based.
Repeated Topics: Conditional probability, multiplication theorem, independent events, total probability theorem, Bayes' theorem (very high repeat), random variables (mean, variance), binomial probability distribution, Bernoulli trials.
Difficulty: Medium; careful use of formulas (especially Bayes' tree diagram) and binomial expansion key. Application questions increased in recent years.
Changes: More Bayes' theorem with real-life scenarios (medical tests, defective items) and binomial distribution in 2023–2025.

Most Important and Repeated Questions

Question Example	Type/Marks	Years Repeated	Notes
A bag contains 4 red and 6 black balls. Two balls are drawn successively without replacement. Find P(both red), P(second red given first black), etc. (conditional probability).	Short Answer (2-3 marks)	2021, 2022, 2023, 2024, 2025	Repeated 5x; Use multiplication theorem or tree diagram; P(both red) = (4/10)×(3/9).
Using Bayes' theorem: A person has 3 bags with different ratios of defective/non-defective bulbs. He picks a bag at random, draws a bulb, it's defective. Find probability it came from bag X.	Long Answer (4-6 marks)	2021 Term 2, 2022, 2023, 2024	Repeated 4x; Tree diagram → P(bag X\|defective) = [P(defective\|bag X) P(bag X)] / P(defective total).
A random variable X has binomial distribution B(n,p). Find P(X=k), mean = np, variance = np(1-p) or probability in specific trials (e.g., 5 coins, P(exactly 3 heads)).	Short/Long Answer (3-4 marks)	2022, 2023, 2024, 2025	Repeated 4x; Binomial formula C(n,k) p^k (1-p)^{n-k}; mean & variance direct.
Assertion: If two events A and B are independent, then P(A\|B) = P(A). Reason: P(A ∩ B) = P(A)P(B).	Assertion-Reason (1 mark)	2023, 2024, 2025	Repeated 3x; Both true, reason explains.
Two events A and B such that P(A) = 0.4, P(B) = 0.5, P(A∪B) = 0.7. Find P(A\|B), P(A∩B), check independence.	Short Answer (2-3 marks)	2021 Term 1, 2022, 2023, 2025	Repeated 4x; P(A∩B) = P(A)+P(B)-P(A∪B); P(A\|B) = P(A∩B)/P(B).
A die is thrown 6 times. Find probability of getting exactly 4 sixes or at least 2 sixes (binomial).	Short Answer (3 marks)	2022, 2023, 2024	Repeated 3x; n=6, p=1/6; P(X=4) = C(6,4) (1/6)^4 (5/6)^2.
MCQ: If P(A) = 0.3, P(B) = 0.4 and A, B independent, then P(A∩B) = ? (a) 0.12 (b) 0.7 (c) 0 (d) 1	MCQ (1 mark)	2021 Term 1, 2023, 2024	Repeated 3x; Answer (a) 0.12 = 0.3 × 0.4.
Find mean and variance of binomial random variable with n=10, p=0.3.	Short Answer (2 marks)	2023, 2025	Repeated 2x; Mean = np = 3, Variance = np(1-p) = 2.1.
Case-based: Given medical test accuracy, prior probability of disease, apply Bayes' theorem to find posterior probability.	Case-Based (4-6 marks)	2023, 2025	Repeated 2x; Use tree or formula; very common real-life application.
Prove that if A and B are independent, then A' and B, A and B', A' and B' are also independent.	Proof (3 marks)	2021 Term 2, 2024	Repeated 2x; Use P(A∩B)=P(A)P(B) → show for complements.

Best Format for Preparation and Answering

Study Format: Memorize Bayes' theorem formula & tree diagram. Practice binomial probabilities & mean/variance. Solve 10–15 PYQs daily (especially Bayes' & conditional).
Answer Format: Bayes': Draw tree → write each branch probability → apply formula. Conditional: Show P(A∩B)/P(B). Binomial: Write C(n,k) p^k (1-p)^{n-k}. Always box final probability.
Tips: Bayes' theorem (with real-life context) is highest repeat & marks — master tree method. Check independence carefully. Time: 10–12 mins for 4–6 mark questions. Very scoring chapter if concepts are clear.

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