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Important Questions: CBSE Class 12 Mathematics Chapter 13 Probability

Probability is a scoring favorite among toppers in CBSE Class 12 Mathematics—8 marks that depend more on clarity of method than computational complexity. Yet every year, avoidable errors in conditional probability and Bayes theorem cost students 3-4 marks. This question bank mirrors the 2024-2025 CBSE pattern, giving you 18 exam-authentic questions with model answers that show exactly what examiners reward and what they penalize.

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Key takeaways

✓Probability carries 8 marks in CBSE Class 12 Mathematics board exam, typically distributed as one 5-mark case study and three 1-mark MCQs.
✓Bayes theorem and conditional probability account for 60-70% of the marks in this chapter across recent papers.
✓The multiplication theorem of probability is tested both directly and as a sub-step in longer word problems involving successive events.
✓Random variables and probability distributions appear in exactly one question per year, usually 3 marks or embedded in case-based questions.
✓CBSE expects complete working with proper probability notation—answers without P(A|B) or P(E) notation often lose method marks.
✓Tree diagrams and contingency tables are not mandatory but can prevent calculation errors in replacement and selection problems.
✓Common mistakes include confusing P(A∩B) with P(A|B), applying Bayes theorem without checking mutual exclusivity, and skipping the verification that probabilities sum to 1.

Chapter Overview and Marks Weightage in CBSE Board Exam

Chapter 13 Probability contributes 8 marks to the 80-mark CBSE Class 12 Mathematics board exam. The 2024-2025 assessment scheme typically includes one 5-mark case-based integrated question, one or two 3-mark or 2-mark short-answer questions, and three 1-mark MCQs in Section A. Conditional probability and Bayes theorem together account for approximately 5-6 marks, while the multiplication theorem appears in 2-3 marks worth of problems. Random variables and probability distributions show up in roughly one question per paper, often woven into the case study. The chapter builds directly on Class 11 probability foundations, so students must be comfortable with basic event algebra, mutually exclusive and independent events, and the addition theorem before tackling conditional probability.

5-mark case-based question: usually a real-world scenario involving successive events, partitions, or Bayes theorem application
2-3 mark questions: direct application of multiplication theorem, conditional probability formula, or Bayes theorem with two or three partitions
1-mark MCQs: formula recall, simple conditional probability, or identification of independent events
Total time allocation recommended: 12-14 minutes for the 5-mark question, 3-4 minutes per 2-3 mark question, 30 seconds per MCQ

1-Mark Questions: MCQ and Very Short Answer

These questions test direct recall of definitions, quick formula application, or concept identification. CBSE sets four MCQs in Section A; three typically come from Probability. Expect questions on the range of conditional probability, recognition of independent events, multiplication theorem application, or checking if events form a partition. Each MCQ carries one mark with no negative marking, but skipping them is wasteful since they require under 30 seconds each. Practice identifying key phrases: 'given that' signals conditional probability, 'without replacement' often means dependent events, and 'mutually exclusive and exhaustive' indicates a partition suitable for Bayes theorem.

2-Mark Questions with Model Answers

Two-mark questions demand one clear application of a theorem or a two-step calculation. Typical formats include computing P(A∩B) using the multiplication theorem, finding conditional probability from a contingency table, or verifying independence of two events. CBSE marking schemes award 1 mark for correct substitution into the formula and 1 mark for the final simplified answer. Always write the formula first—examiners often give the method mark even if arithmetic is wrong. Show fractions in lowest terms and probabilities as decimals only when the question specifies. Keep working concise: two-mark questions should take 3-4 minutes maximum.

3-Mark Questions with Detailed Working

Three-mark questions test multi-step reasoning: applying the multiplication theorem across three events, using Bayes theorem with two partitions, or computing probabilities involving random variables. CBSE typically allocates 1 mark for writing the relevant theorem or formula, 1 mark for correct substitution and intermediate steps, and 1 mark for the final answer. Common question types include bag-and-ball problems with replacement conditions, card-drawing scenarios, and simple reliability or diagnostic test problems. Always define events clearly at the start—write E₁, E₂, A explicitly—and verify that your partition sums to 1 if using Bayes theorem. Three-mark questions should take 5-6 minutes; if you are stuck after 7 minutes, move on and return later.

5-Mark Case-Based and Long-Answer Questions

The 5-mark question in Chapter 13 Probability is almost always a case study: a real-world scenario—medical diagnostics, quality control, survey data, games—with four sub-parts carrying 1+1+1+2 or 1+1+1.5+1.5 marks. Sub-parts (i)-(iii) test direct probability, conditional probability, or independence; sub-part (iv) requires Bayes theorem or law of total probability. CBSE expects students to extract data from a paragraph or table, define events clearly, set up partitions if needed, and show complete working. To maximize marks, underline or list given probabilities, label events E₁, E₂,... and A, write formulas before substituting, and box final answers. Even if you cannot solve part (iv), attempt parts (i)-(iii)—they are often independent and fetch 3 marks. Allocate 12-14 minutes for the full case study.

Read the case paragraph twice; highlight numerical data and keywords like 'given that', 'chosen at random', 'without replacement'.
Define events using standard notation; avoid ambiguous phrases like 'the probability of it'—write P(E₁|A) explicitly.
For Bayes theorem problems, write out the full formula first, then compute the denominator separately to avoid arithmetic errors.
If a sub-part asks 'Are A and B independent?', always check both P(A∩B)=P(A)·P(B) and state the conclusion in words.
Box or underline final answers for each sub-part; examiners skim for these when awarding marks.

Additional 3-Mark and 5-Mark Practice Questions

Mastery in Probability comes from solving diverse problem types until pattern recognition becomes automatic. The questions below cover multiplication theorem with dependent events, partition-based Bayes theorem, and random-variable expectations. Practice writing every step: CBSE awards method marks generously if the approach is correct even when final arithmetic slips. Time yourself—3-mark questions in 5-6 minutes, 5-mark in 12-14 minutes. If you consistently overshoot, you need to streamline notation and skip redundant explanations.

How CBSE Frames Questions from This Chapter

CBSE question-setters follow predictable patterns. Conditional probability questions embed the phrase 'given that' or 'if it is known that'; students must identify the conditioning event and apply P(A|B)=P(A∩B)/P(B). Multiplication theorem questions use scenarios with successive trials—drawing cards, picking marbles, tossing coins repeatedly—and often specify 'with replacement' or 'without replacement' to signal independence or dependence. Bayes theorem problems present a partition of causes (machines, suppliers, disease/no disease) and ask for the probability of a cause given an observed effect; look for phrases like 'it was found that' or 'a randomly chosen item is defective'. Random variables appear in probability distribution tables or as functions of outcomes; expect to compute expectation E(X) or verify ΣP(X=xᵢ)=1. Case studies layer two or three of these concepts, testing whether students can parse real-world context into clean event definitions.

Direct conditional probability: 'Find P(A|B)' with P(A), P(B), P(A∩B) given or derivable from a Venn diagram or table.
Multiplication theorem in succession: 'Two cards drawn without replacement from a deck; find probability both are aces.'
Law of total probability: 'A bag is chosen at random from three bags with different compositions; find probability of drawing a red ball.'
Bayes theorem reverse probability: 'Given the observed outcome, find the probability it came from a specific source.'
Independence verification: 'Check if events A and B are independent' by comparing P(A∩B) with P(A)·P(B).
Random variable expectation: 'A random variable X has distribution... find E(X)' using E(X)=Σxᵢ·P(X=xᵢ).

Common Mistakes Students Make and How to Avoid Them

Five errors account for the majority of mark loss in Probability. First, confusing P(A∩B) with P(A|B)—students write P(A|B) when they mean the joint probability and vice versa. Always ask: am I conditioning on an event (vertical bar) or finding the intersection (cap)? Second, forgetting to verify that a set of events forms a partition before applying Bayes theorem—events must be mutually exclusive and exhaustive. Third, mis-applying the multiplication theorem by treating dependent events as independent; watch for 'without replacement' or any change in sample space between trials. Fourth, arithmetic slips in Bayes denominator—students compute P(E₁)·P(A|E₁) correctly but add the partition terms wrongly. Always write the denominator as a separate line. Fifth, leaving probabilities as complex fractions instead of simplifying or converting to decimals when the question asks; CBSE answer keys show simplified forms, and examiners may deduct for unsimplified final answers.

Mistake 1: Writing P(A|B)=P(A)·P(B). Correct: P(A|B)=P(A∩B)/P(B) unless A and B are independent, in which case P(A|B)=P(A).
Mistake 2: Applying Bayes without checking mutual exclusivity. Example: events 'student plays cricket' and 'student plays football' are not mutually exclusive if overlap exists.
Mistake 3: Using P(A∩B)=P(A)·P(B) when events are dependent. Correct form: P(A∩B)=P(A)·P(B|A).
Mistake 4: Forgetting the law of total probability in the denominator of Bayes. Always expand P(A)=ΣP(Eᵢ)·P(A|Eᵢ).
Mistake 5: Not simplifying fractions or leaving answers as 15/80 instead of 3/16. CBSE marking schemes show reduced forms; match them.

Tips for Scoring Full Marks in Probability

Probability rewards discipline. First, read every question twice and underline 'given that', 'without replacement', 'mutually exclusive', 'independent'—these words dictate which formula to use. Second, always define your events in symbols before writing probabilities; write E₁=event from machine A, A=item is defective. Third, write the formula before substituting numbers—examiners award method marks even if you plug in wrong values. Fourth, for Bayes theorem, compute the denominator on a separate line and label it P(A); this prevents addition errors and makes your working easy to follow. Fifth, verify your final probability is between 0 and 1; if you get 1.2 or −0.3, recheck immediately. Sixth, practice past-year papers under timed conditions; Probability questions are predictable, and pattern recognition saves minutes. Seventh, if stuck, move to the next part or question—Probability sub-parts are often independent, so you can score part (i) even if part (ii) is unclear.

Use tree diagrams for multi-stage problems (drawing balls, card succession); visual clarity reduces errors.
For Bayes, list P(E₁), P(E₂), P(A|E₁), P(A|E₂) in a small table before substituting into the formula.
Check your partition: does P(E₁)+P(E₂)+...=1? If not, you have missed an event or double-counted.
Simplify fractions as a final step, but keep intermediate steps in fraction form to avoid rounding errors.
In case studies, answer every sub-part even if part (iv) is complex—parts (i)-(iii) are often straightforward and fetch 3 marks.

How CBSETUTOR.ai Helps You Master Probability in Less Time

Probability is one of those chapters where a single conceptual gap—mixing up P(A∩B) and P(A|B), or misunderstanding partitions—can cost you 4-5 marks across multiple questions. CBSETUTOR.ai offers a 24×7 AI tutor that you can question as many times as you need, upload a photo of any worked problem for instant step-by-step feedback, and access hundreds of exam-pattern questions with video solutions. Whether you are confused about when to use the multiplication theorem versus Bayes, or you keep making arithmetic errors in multi-part case studies, the AI tutor identifies your mistake pattern and drills you on exactly those weak spots. Parents across Mumbai, Pune, Bengaluru, and NCR trust CBSETUTOR.ai because it delivers clarity fast—no waiting for weekend tuition, no expensive per-hour fees. At a flat ₹999 per month for any class from 6 to 12, your child gets unlimited doubt solving, chapter-wise test series, and progress tracking. Start with a free 3-day trial and watch your Probability scores climb from 5/8 to 8/8 within two weeks of focused practice.

Upload a photo of your Bayes theorem working; the AI tutor highlights exactly where you went wrong—denominator error, wrong partition, or notation mix-up.
Get auto-generated practice sets that mirror CBSE case-study patterns, so you never face a surprise format in the board exam.
Watch concise video solutions for every NCERT exercise and past-year question, narrated in clear Indian English.
Track your accuracy and speed on 1-mark MCQs versus 5-mark case studies; the dashboard shows you where to focus your last-minute revision.

Revision Strategy: Last 15 Days Before the Board Exam

If the board exam is two weeks away, prioritize high-yield topics. Day 1-3: drill conditional probability and multiplication theorem—solve 10 varied problems daily until P(A|B) versus P(A∩B) becomes automatic. Day 4-6: master Bayes theorem with three-partition problems; practice writing the full formula and computing the denominator separately. Day 7-9: solve past five years' 5-mark case studies under timed conditions (12 minutes each); analyze marking schemes to see where method marks are awarded. Day 10-12: focus on 1-mark MCQs—speed matters; aim for 30 seconds per question. Day 13-14: take one full-length sample paper, mark it honestly, and redo every Probability question you got wrong. Day 15: light revision—rewrite key formulas, scan your error log, and sleep well. Do not attempt new tough problems the night before; confidence matters as much as knowledge.

Formula sheet: write P(A|B)=P(A∩B)/P(B), P(A∩B)=P(A)·P(B|A), Bayes P(Eᵢ|A)=[P(Eᵢ)·P(A|Eᵢ)]/[ΣP(Eⱼ)·P(A|Eⱼ)], E(X)=Σxᵢ·P(xᵢ) on one page.
Error log: list every mistake (e.g. 'forgot to check partition sums to 1', 'used P(A)·P(B) for dependent events') and review it daily.
MCQ blitz: solve 20 MCQs in 10 minutes every morning; this builds speed and pattern recognition.
Case-study walkthrough: pick three past-year case studies, solve them, then watch solution videos to compare your method with the examiner's expected approach.

Frequently asked questions

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

Probability carries 8 marks, typically one 5-mark case-based question, one or two 2-3 mark short-answer questions, and three 1-mark MCQs in Section A.

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

P(A∩B) is the probability both A and B occur together (joint probability). P(A|B) is the probability of A occurring given that B has already occurred (conditional probability). Formula: P(A|B)=P(A∩B)/P(B).

When should I use Bayes theorem instead of the multiplication theorem?+

Use Bayes theorem when you know the probability of an effect given various causes and need to find the probability of a specific cause given the observed effect. Use multiplication theorem when computing the probability of successive events in a known order.

How do I know if two events are independent?+

Events A and B are independent if P(A∩B)=P(A)·P(B), or equivalently if P(A|B)=P(A). Check by computing both sides; if they are equal, the events are independent.

What is a partition of the sample space and why does it matter for Bayes theorem?+

A partition is a collection of mutually exclusive and exhaustive events E₁, E₂,... such that exactly one occurs in every trial. Bayes theorem requires a partition because the denominator P(A)=ΣP(Eᵢ)·P(A|Eᵢ) sums over all possible causes.

Do I lose marks if I do not simplify fractions in my final answer?+

CBSE marking schemes show simplified fractions or decimals. While you may not lose a full mark, examiners prefer clean final answers. Always reduce fractions to lowest terms unless the question specifies 'leave in fraction form'.

How much time should I spend on the 5-mark Probability case study in the board exam?+

Allocate 12-14 minutes maximum. If you are stuck on part (iv) after 10 minutes, move on and return later—parts (i)-(iii) often fetch 3 marks and are easier.

Is it necessary to draw tree diagrams for multi-stage probability problems?+

Not mandatory, but tree diagrams prevent errors in problems involving successive events without replacement or multiple stages. If you are comfortable with algebraic setup, skip the diagram to save time.

What are the most common mistakes in Bayes theorem questions?+

Forgetting to compute the full denominator P(A)=ΣP(Eᵢ)·P(A|Eᵢ), using P(A)·P(B) for dependent events, and not verifying that the partition events are mutually exclusive and exhaustive.

How can CBSETUTOR.ai help if I keep making errors in conditional probability?+

Upload a photo of your worked problem to the AI tutor; it identifies whether you confused P(A|B) with P(A∩B), applied the wrong formula, or made an arithmetic slip. You get instant feedback and similar practice problems until the concept clicks. Try the free 3-day trial at ₹999/month for unlimited doubt solving.

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