The NCERT Solutions for Class 11 Maths Chapter 14 Probability Miscellaneous Exercise solve every question of the last exercise, according to the latest 2026-27 CBSE syllabus. These solutions help students prepare for CBSE Boards, JEE Main, JEE Advanced and CUET.
Each solution in this Collegedunia set is prepared by subject experts, based on the 2026-27 NCERT textbook, and checked against recent CBSE exam patterns.
The Miscellaneous Exercise mixes every idea of the chapter into higher-order problems. It brings together these skills:
- Counting outcomes with combinations in card and committee problems.
- Using the addition rule and complement rule together in one question.
- Reading odds in favour and odds against to find a probability.
Topics Covered in Probability Miscellaneous Exercise
This exercise is a mixed test of the whole chapter. It pulls in sample space, events and the axiomatic rules, and often needs permutations and combinations to count outcomes. The questions are longer than the two main exercises and reward a clear, step-by-step method. Students meet card draws, letter arrangements, dice and committee selections in one place.
- Combination counting: using nCr to count favourable and total outcomes.
- Addition and complement rules: combining P(A ∪ B) with P(A′) in a single problem.
- Odds: converting odds in favour a : b into a probability a/(a + b).
- Three-event problems: extending the addition rule to P(A ∪ B ∪ C).
- Word problems: reading a real situation and choosing the right rule.
Important: most Miscellaneous questions need counting with combinations before any probability rule is applied. Getting n(S) and n(A) right is half the solution.
Question-wise Breakdown of Probability Miscellaneous Exercise
The NCERT Solutions for Class 11 Maths Chapter 14 Probability Miscellaneous Exercise work through all 10 questions in order. The table below groups the questions by type so students can plan their practice. Full step-by-step answers are inside the PDF above.
| Questions | What they ask | Skill tested |
|---|---|---|
| Q1 to Q3 | Probability using combinations for selecting objects | Counting with nCr |
| Q4 to Q6 | Odds in favour and against, and their probabilities | Odds to probability |
| Q7 to Q8 | Addition rule and complement rule combined | P(A ∪ B) and P(A′) |
| Q9 to Q10 | Longer word problems on letters, dice and cards | Applying all rules together |
These questions match the higher-mark board questions and the entrance-exam style, so they are worth practising fully.
Key Formulas and Definitions for Probability Miscellaneous Exercise
The Miscellaneous Exercise uses every result of the chapter plus counting tools from Permutations and Combinations. The table lists the formulas students need most here.
| Formula / Rule | Meaning |
|---|---|
| P(A) = n(A)/n(S) | Probability with equally likely outcomes |
| nCr = n! / [r!(n − r)!] | Number of ways to choose r objects from n |
| P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | Addition rule for two events |
| P(A′) = 1 − P(A) | Complement rule |
| Odds in favour a : b | Probability of the event = a/(a + b) |
| Odds against a : b | Probability of the event = b/(a + b) |
| P(A ∪ B ∪ C) | Sum of singles − sum of pairs + P(A ∩ B ∩ C) |
Tip: when a question gives odds instead of probability, convert to a fraction first, then apply the usual rules. Odds in favour of 3 : 2 means a probability of 3/5.
Common Mistakes in Probability Miscellaneous Exercise
Students lose marks in the Miscellaneous Exercise on counting and on reading the question. The list below covers the errors that show up most often in tests on the NCERT Solutions for Class 11 Maths Chapter 14 Probability Miscellaneous Exercise.
- Using permutations when the question needs combinations. Selecting a committee is order-free, so use nCr.
- Mixing up odds in favour with odds against. For odds in favour a : b the probability is a/(a + b), not b/(a + b).
- Dropping a term in the three-event addition rule, which needs the pairwise and triple intersections.
- Not simplifying the final fraction. A probability answer should be written in lowest terms.
Practice Questions for Probability Miscellaneous Exercise
After reading the solutions, students should test themselves on the same question types. The card below opens a set of solved practice questions with step-by-step answers for the Miscellaneous Exercise.
Practice Card: Solved Practice Questions for Class 11 Maths Probability Miscellaneous Exercise — attempt each question, then check the worked solution.
Other Exercises of Chapter 14 Probability
Chapter 14 Probability has two exercises plus this Miscellaneous Exercise. Use the table to revise the earlier exercises alongside the mixed questions.
| Exercise | NCERT Solutions link |
|---|---|
| Exercise 14.1 | Class 11 Maths Probability Exercise 14.1 Solutions |
| Exercise 14.2 | Class 11 Maths Probability Exercise 14.2 Solutions |
More Class 11 Maths Probability Resources
Pair the solutions with the notes and the NCERT book PDF for full chapter revision. All three resources for Chapter 14 Probability are linked below.
| Resource | Link |
|---|---|
| Revision Notes | Class 11 Maths Probability Notes |
| Handwritten Notes | Class 11 Maths Probability Handwritten Notes |
| NCERT Book PDF | Class 11 Maths Probability NCERT Book PDF |
NCERT Solutions for Other Class 11 Maths Chapters
Students can move to the first exercise of any other Class 11 Maths chapter using the cross-sell table below.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1: Sets | Sets Solutions |
| Chapter 2: Relations and Functions | Relations and Functions Solutions |
| Chapter 3: Trigonometric Functions | Trigonometric Functions Solutions |
| Chapter 4: Complex Numbers and Quadratic Equations | Complex Numbers Solutions |
| Chapter 5: Linear Inequalities | Linear Inequalities Solutions |
| Chapter 6: Permutations and Combinations | Permutations and Combinations Solutions |
| Chapter 7: Binomial Theorem | Binomial Theorem Solutions |
| Chapter 8: Sequences and Series | Sequences and Series Solutions |
| Chapter 9: Straight Lines | Straight Lines Solutions |
| Chapter 10: Conic Sections | Conic Sections Solutions |
| Chapter 11: Introduction to Three Dimensional Geometry | Three Dimensional Geometry Solutions |
| Chapter 12: Limits and Derivatives | Limits and Derivatives Solutions |
| Chapter 13: Statistics | Statistics Solutions |
| Chapter 14: Probability | You are here |
Student Feedback
In a Collegedunia survey of 11,970 Class 11 students conducted before the 2026 exams, 69% of students said the combination-based counting in the Miscellaneous Exercise was the toughest step, while converting odds to a probability was rated the quickest skill to master.
FAQs on NCERT Solutions for Class 11 Maths Chapter 14 Probability Miscellaneous Exercise
Probability NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Class 11 Maths Chapter 14 Probability Miscellaneous Exercise?
Ans. The Miscellaneous Exercise has 10 questions. They mix combination counting, the addition and complement rules, odds in favour and against, and longer word problems on cards, dice, letters and committees. They are higher-order questions that test the whole chapter together.
Ques. How do you convert odds in favour into a probability?
Ans. If the odds in favour of an event are a : b, then the probability of the event is a/(a + b). For example, odds in favour of 3 : 2 give a probability of 3/5. For odds against a : b, the probability of the event is b/(a + b).
Ques. Why are combinations used in the Miscellaneous Exercise?
Ans. Many questions select a group of objects, like cards from a deck or members of a committee, where order does not matter. The number of such selections is nCr. Both the total outcomes n(S) and the favourable outcomes n(A) are counted this way before applying P(A) = n(A)/n(S).
Ques. How do you find the probability of A or B for three events?
Ans. For three events, P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(B ∩ C) − P(A ∩ C) + P(A ∩ B ∩ C). Missing any of the pairwise or triple terms is a common cause of lost marks in the Miscellaneous Exercise.
Ques. Are the NCERT Solutions for Class 11 Maths Chapter 14 Probability Miscellaneous Exercise free to download?
Ans. Yes. The NCERT Solutions for Class 11 Maths Chapter 14 Probability Miscellaneous Exercise are free to download as a PDF from this page. Every question is solved step by step, based on the 2026-27 NCERT textbook, so students can revise offline for CBSE Boards, JEE Main, JEE Advanced and CUET.
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