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CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability Worksheet with Answers

Probability is one of the most practical chapters in Class 9 Mathematics, equipping you to analyse uncertain events—from dice rolls to weather forecasts. This worksheet covers NCERT Chapter 7 comprehensively: experimental versus theoretical probability, sample spaces, events, and real-world statistical probability. Structured to mirror the latest CBSE exam format, it progresses from quick MCQs to challenging HOTS questions, ending with a case study. Set a timer for 90 minutes, attempt all sections honestly, then check the detailed answer key to identify gaps and solidify understanding.

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Key takeaways

✓Experimental probability uses real trial data; theoretical probability assumes equally likely outcomes—both are covered in this worksheet.
✓Sample space clarity is critical: every MCQ and short-answer question tests your ability to list all possible outcomes correctly.
✓The worksheet includes three HOTS questions demanding tree diagrams, multi-step reasoning, and real-world application of probability concepts.
✓A dedicated case-study problem mirrors the latest CBSE pattern, integrating data interpretation with probability calculation.
✓Complete answer key with step-by-step explanations ensures self-learners can verify logic, not just final answers.
✓Suggested time is 90 minutes; difficulty ranges from foundational (MCQs) to advanced (HOTS and case study) to match term exam rigor.
✓Perfect for quick revision or diagnostic practice before CBSE Class 9 term exams—download, print, and solve offline.

Quick Chapter Recap: Introduction to Probability

A random experiment is any action where you know all possible outcomes in advance but cannot predict which one will occur in a single trial—tossing a coin, rolling a die, or drawing a card. The sample space (S) lists every possible outcome; for example, rolling a standard die gives S = {1, 2, 3, 4, 5, 6} with n(S) = 6. An event is any subset of the sample space; rolling an even number is the event E = {2, 4, 6}. Experimental (empirical) probability is calculated from actual trials: divide the number of times the event occurred by the total number of trials. Theoretical probability assumes all outcomes are equally likely and uses the formula P(E) = (Number of favourable outcomes) / (Total number of outcomes). The probability scale runs from 0 (impossible) to 1 (certain), with 0.5 representing equal likelihood. Tree diagrams help visualise multi-step experiments and count outcomes systematically. Real-world applications use statistical probability: collect sample data to estimate population probabilities, a technique used in surveys, quality control, and forecasting.

Random experiment: unpredictable outcome from known possibilities
Sample space S: complete list of all outcomes; n(S) is its size
Event E: subset of S; favourable outcomes belong to E
Experimental probability = (times event occurred) / (total trials)
Theoretical probability = (favourable outcomes) / (total outcomes) — only when equally likely
Probability always lies between 0 and 1 inclusive
Tree diagrams systematically map multi-step experiments

Section A: Multiple Choice Questions (MCQs)

This section contains six multiple-choice questions testing foundational understanding of probability definitions, sample spaces, and quick calculations. Each question carries 1 mark. Choose the single best answer. These MCQs mirror the objective-type questions that appeared in the CBSE 2024–25 term exams and are designed to check clarity on experimental versus theoretical probability, the concept of equally likely outcomes, and interpretation of probability values. Read each option carefully; distractors are common student errors. Aim to complete this section in 10–12 minutes.

Q1. A coin is tossed once. What is the probability of getting a tail? (a) 0 (b) 0.25 (c) 0.5 (d) 1
Q2. A die is rolled. What is P(getting a number greater than 6)? (a) 0 (b) 1/6 (c) 1 (d) undefined
Q3. In 50 tosses of a coin, heads appeared 28 times. The experimental probability of heads is: (a) 0.28 (b) 0.5 (c) 0.56 (d) 0.72
Q4. Two coins are tossed together. The sample space contains how many outcomes? (a) 2 (b) 3 (c) 4 (d) 8
Q5. Which event has probability equal to 1? (a) Getting a 7 on a standard die (b) Getting a number less than 10 on a standard die (c) Getting heads on a coin toss (d) Drawing a red card from a deck
Q6. If P(E) = 0.85, the event E is: (a) impossible (b) unlikely (c) equally likely (d) very likely

Section B: Fill in the Blanks

Complete each sentence with the correct term or number. This section tests recall of key definitions and formulae from NCERT Class 9 Mathematics Chapter 7. Each blank carries 1 mark. Write your answers precisely—spelling and terminology matter. Aim to finish this section in 8–10 minutes. These questions ensure you can articulate fundamental concepts in your own exam answers, especially for 1- or 2-mark direct questions that frequently appear in CBSE papers.

Q7. The set of all possible outcomes of a random experiment is called the __________ __________.
Q8. The probability of an impossible event is __________.
Q9. Experimental probability is also known as __________ probability.
Q10. If a die is rolled, the probability of getting an even number is __________ (express as a fraction).
Q11. The sum of probabilities of all outcomes in a sample space is always equal to __________.
Q12. A probability value must lie between __________ and __________ inclusive.

Section C: True or False Statements

Read each statement carefully and write True or False. If false, you may briefly correct the statement in brackets for self-learning. Each question carries 1 mark. This section checks conceptual clarity and common misconceptions—students often confuse experimental with theoretical probability, or assume all experiments have equally likely outcomes. Allocate 6–8 minutes. These true/false items are similar to assertion-reason or short objective questions in recent CBSE board papers, so mastering them builds confidence for quick-mark questions.

Q13. The probability of any event can be greater than 1.
Q14. Theoretical probability assumes all outcomes are equally likely.
Q15. If you toss a fair coin 100 times and get 53 heads, the theoretical probability of heads changes to 0.53.
Q16. The sample space for rolling two dice contains 36 outcomes.
Q17. An event that is certain to happen has probability 0.
Q18. Experimental probability becomes more reliable as the number of trials increases.

Section D: Short Answer Questions (2 or 3 marks each)

Answer each question in 2–4 sentences or show brief working. Each question is worth 2 or 3 marks as indicated. This section mirrors the standard short-answer format in CBSE Class 9 term exams. Examiners look for clear reasoning, correct identification of sample space, accurate calculation, and proper use of probability notation. You should spend roughly 25–30 minutes on this section. Show all steps: define the event, count favourable outcomes, state the formula, substitute, and simplify. Partial credit is awarded for method even if the final answer has a small error, so never skip the working.

Q19. (2 marks) A bag contains 5 red balls and 7 blue balls. One ball is drawn at random. Find the probability that it is blue.
Q20. (2 marks) A die is thrown once. What is the probability of getting a prime number?
Q21. (3 marks) Two coins are tossed simultaneously. List the sample space and find the probability of getting exactly one head.
Q22. (3 marks) In 80 trials of a random experiment, event A occurred 36 times. Calculate the experimental probability of A. If the experiment is conducted 200 times under identical conditions, estimate how many times A is likely to occur.
Q23. (3 marks) A card is drawn from a well-shuffled standard deck of 52 cards. Find the probability that it is (i) a king, (ii) a red card.

Section E: Long Answer and HOTS Questions (4 or 5 marks each)

These three questions demand higher-order thinking: multi-step reasoning, construction of tree diagrams, or application of probability to unfamiliar real-world contexts. Each carries 4 or 5 marks. Allocate 20–25 minutes. CBSE values clear diagrams, logical flow, and complete explanations. For tree diagrams, label every branch and outcome. For word problems, define events explicitly, show all arithmetic, and state conclusions in context. These mirror the challenging 4–5 mark questions that often decide A1 versus A2 grades in board exams, so practice thoroughly.

Q24. (4 marks) A coin is tossed three times. (a) Draw a tree diagram showing all possible outcomes. (b) List the sample space. (c) Find the probability of getting at least two heads.
Q25. (5 marks) A bag contains 3 red, 4 green, and 5 yellow balls. A ball is drawn at random, its colour is noted, and it is replaced. Then a second ball is drawn. (a) How many total outcomes are possible? (b) Find the probability that both balls are green. (c) Find the probability that the two balls are of different colours.
Q26. (5 marks) A manufacturer tested 500 light bulbs. 460 lasted more than 1000 hours. (a) Find the experimental probability that a bulb lasts more than 1000 hours. (b) If the company produces 50,000 bulbs, estimate how many will last more than 1000 hours. (c) Explain why this estimate might differ from the actual count and what the manufacturer could do to improve accuracy.

Section F: Case Study Question (4 marks)

Read the passage below carefully, study the data, and answer the four sub-questions. Case-study questions were introduced in the CBSE 2020–21 session and now appear regularly in Class 9 term exams, carrying 4 marks total (usually 1+1+1+1 or 1+1+2). They test your ability to extract information, apply probability formulae, and interpret results in context. Spend 10–12 minutes. Show working for calculation-based sub-parts. This format trains you to handle real-world data—exactly the skill tested in board exams and useful in projects, surveys, and everyday decision-making.

Complete Answer Key with Explanations

Below are the correct answers for every question, along with brief working or reasoning. Use this key to verify your responses and understand where you went wrong. For MCQs and fill-in-the-blanks, check your answer matches exactly. For short and long answers, compare your method and logic—CBSE awards marks for correct steps even if the final answer has a minor slip. Self-assessment is most effective when you write down why you made an error (concept gap, calculation mistake, misread question) and revisit that topic in your NCERT Class 9 Mathematics textbook or notes. If you scored below 70 percent, focus on the quick recap and reattempt weak sections after a day. For persistent doubts, platforms like CBSETUTOR.ai offer 24×7 AI tutor support with photo-upload solving at ₹999/month flat across all classes (6–12), plus a 3-day free trial—perfect for instant doubt clearing without waiting for the next tuition class.

Section A Answers: Q1(c) 0.5, Q2(a) 0, Q3(c) 0.56, Q4(c) 4, Q5(b) certain—all outcomes <10, Q6(d) very likely
Section B Answers: Q7 sample space, Q8 0, Q9 empirical, Q10 1/2 or 3/6, Q11 1, Q12 0 and 1
Section C Answers: Q13 False (max is 1), Q14 True, Q15 False (theoretical stays 0.5; experimental is 0.53), Q16 True, Q17 False (certain = 1), Q18 True
Q19: P(blue)=7/12. Q20: Primes {2,3,5}→3 outcomes, P=3/6=1/2. Q21: S={HH,HT,TH,TT}; exactly one head={HT,TH}→2/4=1/2. Q22: P(A)=36/80=0.45; in 200 trials≈0.45×200=90. Q23: (i) 4/52=1/13, (ii) 26/52=1/2.
Q24: (a) Tree: 1st toss H/T, 2nd H/T, 3rd H/T→8 branches. (b) S={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}. (c) ≥2H={HHH,HHT,HTH,THH}→4/8=1/2.
Q25: (a) 12×12=144. (b) P(GG)=4/12×4/12=16/144=1/9. (c) P(same colour)=(3²+4²+5²)/144=50/144; P(different)=1−50/144=94/144=47/72.
Q26: (a) 460/500=0.92. (b) 0.92×50000=46000 bulbs. (c) Sample variation; increase sample size or test multiple batches.
Case Study: (i) 80/200=2/5=0.4. (ii) 1−60/200=140/200=7/10=0.7. (iii) 40/200×1000=200. (iv) Experimental—based on actual survey data, not assumption of equal likelihood.

How to Use This Worksheet Effectively

Print this worksheet and attempt it in one sitting under exam conditions: 90 minutes, no textbook, no phone. Write all working on separate sheets as you would in a board exam—this trains speed and neatness. After time is up, stop and mark your paper using the answer key, awarding partial credit for correct steps even if the final answer is wrong. Calculate your percentage: aim for 80 percent or higher to be confident for term exams. Identify patterns in your errors: Did you miscount sample spaces? Confuse experimental with theoretical probability? Make arithmetic slips? Revisit those NCERT sections and reattempt similar questions from your Class 9 Mathematics textbook exercises. Use this worksheet weekly during revision month—repetition with variation builds speed and accuracy. For instant doubt resolution, upload tricky questions to CBSETUTOR.ai and get step-by-step solutions from the AI tutor any time, day or night.

Set a 90-minute timer and work in silence, simulating exam pressure
Show full working for every question—examiners award method marks generously in CBSE
Self-mark using the answer key, then calculate section-wise scores to spot weak areas
Review incorrect answers: read the explanation, redo the question, then try a similar NCERT exercise problem
Reattempt the worksheet after one week—aim to improve your score by at least 10 percent
Combine this with NCERT exemplar problems and previous years' CBSE question papers for comprehensive practice

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Common Mistakes to Avoid in Probability Questions

Students frequently lose marks on probability problems not because they lack understanding, but because of small, preventable errors. One classic mistake is incomplete sample space: when tossing two coins, writing {HH, HT, TT} and forgetting TH. Always list outcomes systematically—draw a tree if needed. Another error is confusing experimental and theoretical probability: theoretical assumes equal likelihood and uses logic; experimental uses actual trial data. Do not mix them. Many students write probabilities greater than 1 or negative, which is impossible—always check 0 ≤ P ≤ 1. In replacement problems, students forget that replacing a ball means the second draw is independent with the same sample space; without replacement, the sample space shrinks. Read the question twice. For case studies, underline key numbers and check units (out of 200? out of 1000?). Finally, in exams, always simplify fractions (write 1/2, not 3/6) and express probabilities as fractions unless the question asks for decimals. These small habits can lift your score by 10–15 percent without learning any new concepts.

Incomplete sample space—list all outcomes methodically, use tree diagrams for multi-step experiments
Mixing experimental and theoretical: experimental = real data / trials; theoretical = favourable / total equally likely outcomes
Probability outside [0,1]—any answer <0 or >1 is wrong; double-check your arithmetic
Ignoring replacement: 'with replacement' means independent draws; 'without replacement' shrinks the sample space
Not simplifying fractions: 2/4 should be 1/2; examiners may deduct marks for unsimplified answers
Misreading case-study numbers: underline data, check denominators, ensure your calculation matches the question context

Frequently asked questions

What is the difference between experimental and theoretical probability?+

Experimental probability is calculated from actual trial data: divide the number of times an event occurred by the total number of trials performed. Theoretical probability assumes all outcomes are equally likely and uses the formula (favourable outcomes)/(total outcomes). Experimental values vary with sample size and approach theoretical values as trials increase, according to the Law of Large Numbers.

How do I know if outcomes are equally likely?+

Outcomes are equally likely when the experiment is fair and symmetric—like a balanced coin, unbiased die, or well-shuffled deck. If the question states 'fair', 'unbiased', or 'well-shuffled', assume equal likelihood and use theoretical probability. If fairness is uncertain or the experiment is real-world (e.g. weather, human behaviour), use experimental or statistical probability.

Why does my experimental probability not match the theoretical value?+

Small sample sizes produce random fluctuations. For example, tossing a fair coin 10 times might yield 7 heads (0.7 experimental) versus 0.5 theoretical. As you increase trials to 100, 1000, or more, experimental probability converges toward the theoretical value. This is normal and expected—it is why large samples are more reliable.

How many questions on probability appear in the CBSE Class 9 Maths term exam?+

Typically, Chapter 7 contributes 6–10 marks in the term exam: 1–2 MCQs (1 mark each), 1–2 short-answer questions (2–3 marks), one long-answer or HOTS question (4–5 marks), and often part of a case-study question (1–2 marks). Exact distribution varies yearly, so practice all question types thoroughly.

What is a sample space and why is listing it important?+

The sample space S is the complete set of all possible outcomes of a random experiment. Listing it correctly ensures you count total outcomes accurately, which is the denominator in P(E)=n(E)/n(S). An incomplete or incorrect sample space leads to wrong probabilities. Use tree diagrams or systematic listing (e.g. lexicographic order) to avoid missing outcomes.

Can probability ever be greater than 1 or less than 0?+

No. By definition, probability measures the fraction of outcomes favouring an event, so 0 ≤ P(E) ≤ 1 always. P=0 means impossible; P=1 means certain. If your calculation gives a value outside this range, recheck your sample space count and favourable outcomes—there is an error somewhere.

How do I draw a tree diagram for tossing three coins?+

Start with a single point (root). Draw two branches labelled H and T for the first toss. From each of those, draw two branches (H, T) for the second toss, giving four paths. From each of those four, draw two branches for the third toss, yielding eight end-points. Label each path: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Count: 2×2×2=8 outcomes.

What does 'with replacement' mean in probability questions?+

With replacement means after you draw an item (e.g. a ball from a bag), you return it before the next draw. This keeps the sample space and probabilities unchanged for each draw. Without replacement, the item is not returned, so the sample space shrinks and probabilities change. Always read the question carefully to identify which scenario applies.

How can I estimate outcomes for a large population using probability?+

Calculate the probability from your sample, then multiply by the population size. For example, if 40 out of 200 students prefer cricket, P(cricket)=40/200=0.2. For 1000 students, estimate 0.2×1000=200 prefer cricket. This is statistical probability—widely used in surveys, quality control, and forecasting.

Where can I find more probability practice questions beyond this worksheet?+

Refer to NCERT Class 9 Mathematics textbook Exercise 15.1 (renamed Exercise 7.1 in some editions), NCERT Exemplar, and previous years' CBSE sample papers. For instant, unlimited practice with step-by-step solutions, try CBSETUTOR.ai—upload any doubt photo 24×7 and get explanations immediately. The 3-day free trial at ₹999/month flat across all classes is ideal for exam prep.

Related resources

CBSE Class 9 Mathematics Chapter 7 Triangles Worksheet with Answers Important Questions: CBSE Class 9 Mathematics Chapter 7 Triangles Class 9 Mathematics Chapter 7 Triangles — Formulas & Key Points Important Questions: CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability CBSE Class 9 Mathematics Chapter 1 Number Systems — Notes CBSE Class 9 Mathematics — Number Systems: complete chapter guide NCERT Solutions for Class 9 Mathematics Chapter 1: Number Systems – Complete Guide CBSE Class 9 Mathematics Chapter 1 Number Systems Worksheet with Answers

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