- Probability expresses how likely an event is, from 0 (impossible) to 1 (certain).
- A random experiment has known possible outcomes but an uncertain individual result.
- The sample space is the set of all possible outcomes of an experiment.
- An event is a subset of the sample space.
- The classical formula applies when elementary outcomes are equally likely.
- For finite equally likely outcomes, P(E)=number favourable/number total.
- P(not E)=1−P(E).
- An impossible event has probability 0 and a certain event probability 1.
- Estimate probability by relative frequency in repeated trials.
- Theoretical probability is model-based; experimental estimates vary due to chance and approach stable values over many trials under suitable conditions.
- For two fair independent coins, ordered outcomes are HH, HT, TH, TT.
- Two independent fair dice have 36 equally likely ordered outcomes.
Use these notes alongside your current textbook. Check important formulas, worked answers and syllabus coverage with your teacher or official learning material.
The Mathematics of Maybe: Introduction to Probability — Class 9 Mathematics
Textbook edition: Ganita Manjari 2026–27, Part I, Chapter 7. This chapter belongs to the 2026–27 Ganita Manjari sequence.
Overview: Probability expresses how likely an event is, from 0 (impossible) to 1 (certain).
This is an original expanded study guide. Exact numbered NCERT exercises, figures and activities have not been independently checked against the complete official textbook; the practice below is supplementary and is not mislabelled as official exercise solutions.
Learning outcomes
- Understand definitions and necessary conditions.
- Use mathematical notation and accurate diagrams.
- Apply theorems and formulas with worked reasoning.
- Identify errors and check numerical results.
Uncertainty and chance
Probability expresses how likely an event is, from 0 (impossible) to 1 (certain).
Worked example
A fair coin has probability 1/2 of heads.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Random experiment
A random experiment has known possible outcomes but an uncertain individual result.
Worked example
Rolling a fair six-sided die.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Sample space
The sample space is the set of all possible outcomes of an experiment.
Worked example
One coin toss: S={H,T}.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Events
An event is a subset of the sample space.
Worked example
Rolling an even number: E={2,4,6}.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Equally likely outcomes
The classical formula applies when elementary outcomes are equally likely.
Worked example
For a fair die, each face has probability 1/6.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Classical probability
For finite equally likely outcomes, P(E)=number favourable/number total.
Worked example
P(odd on fair die)=3/6=1/2.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Complementary events
P(not E)=1−P(E).
Worked example
If P(rain)=0.3, P(no rain)=0.7.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Impossible and certain events
An impossible event has probability 0 and a certain event probability 1.
Worked example
Rolling 7 on a standard six-sided die has probability 0.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Empirical probability
Estimate probability by relative frequency in repeated trials.
Worked example
If heads appears 48 times in 100 tosses, experimental probability=0.48.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Theoretical versus experimental
Theoretical probability is model-based; experimental estimates vary due to chance and approach stable values over many trials under suitable conditions.
Worked example
50 tosses need not produce exactly 25 heads.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Two coins
For two fair independent coins, ordered outcomes are HH, HT, TH, TT.
Worked example
P(exactly one head)=2/4=1/2.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Two dice
Two independent fair dice have 36 equally likely ordered outcomes.
Worked example
P(sum 7)=6/36=1/6.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Without replacement
When objects are drawn without replacement, later probabilities depend on earlier draws.
Worked example
Bag 3 red,2 blue: P(red then red)=3/5×2/4=3/10.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Independent events
For independent events A,B, P(A and B)=P(A)P(B).
Worked example
Two fair coin heads: 1/2×1/2=1/4.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Mutually exclusive events
Events that cannot happen together have P(A∩B)=0; P(A or B)=P(A)+P(B).
Worked example
On one die, P(1 or 6)=2/6.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
General addition rule
P(A∪B)=P(A)+P(B)−P(A∩B).
Worked example
On a die, A=even (3/6), B=>4 (2/6), intersection {6} (1/6), union=4/6.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Probability misconceptions
Past independent outcomes do not change the next fair toss; do not assume real-world outcomes equally likely without justification.
Worked example
After five heads, next fair toss still has P(heads)=1/2.
Check your understanding
Explain why this example follows the rule and state the assumptions required.
Concept and formula reference
| Concept | Explanation | Worked illustration |
|---|---|---|
| Uncertainty and chance | Probability expresses how likely an event is, from 0 (impossible) to 1 (certain). | A fair coin has probability 1/2 of heads. |
| Random experiment | A random experiment has known possible outcomes but an uncertain individual result. | Rolling a fair six-sided die. |
| Sample space | The sample space is the set of all possible outcomes of an experiment. | One coin toss: S={H,T}. |
| Events | An event is a subset of the sample space. | Rolling an even number: E={2,4,6}. |
| Equally likely outcomes | The classical formula applies when elementary outcomes are equally likely. | For a fair die, each face has probability 1/6. |
| Classical probability | For finite equally likely outcomes, P(E)=number favourable/number total. | P(odd on fair die)=3/6=1/2. |
| Complementary events | P(not E)=1−P(E). | If P(rain)=0.3, P(no rain)=0.7. |
| Impossible and certain events | An impossible event has probability 0 and a certain event probability 1. | Rolling 7 on a standard six-sided die has probability 0. |
| Empirical probability | Estimate probability by relative frequency in repeated trials. | If heads appears 48 times in 100 tosses, experimental probability=0.48. |
| Theoretical versus experimental | Theoretical probability is model-based; experimental estimates vary due to chance and approach stable values over many trials under suitable conditions. | 50 tosses need not produce exactly 25 heads. |
| Two coins | For two fair independent coins, ordered outcomes are HH, HT, TH, TT. | P(exactly one head)=2/4=1/2. |
| Two dice | Two independent fair dice have 36 equally likely ordered outcomes. | P(sum 7)=6/36=1/6. |
| Without replacement | When objects are drawn without replacement, later probabilities depend on earlier draws. | Bag 3 red,2 blue: P(red then red)=3/5×2/4=3/10. |
| Independent events | For independent events A,B, P(A and B)=P(A)P(B). | Two fair coin heads: 1/2×1/2=1/4. |
| Mutually exclusive events | Events that cannot happen together have P(A∩B)=0; P(A or B)=P(A)+P(B). | On one die, P(1 or 6)=2/6. |
| General addition rule | P(A∪B)=P(A)+P(B)−P(A∩B). | On a die, A=even (3/6), B=>4 (2/6), intersection {6} (1/6), union=4/6. |
| Probability misconceptions | Past independent outcomes do not change the next fair toss; do not assume real-world outcomes equally likely without justification. | After five heads, next fair toss still has P(heads)=1/2. |
Exercise A — Core concepts
- Define uncertainty and chance, state the conditions and construct a correct example.
- Define random experiment, state the conditions and construct a correct example.
- Define sample space, state the conditions and construct a correct example.
- Define events, state the conditions and construct a correct example.
- Define equally likely outcomes, state the conditions and construct a correct example.
- Define classical probability, state the conditions and construct a correct example.
- Define complementary events, state the conditions and construct a correct example.
- Define impossible and certain events, state the conditions and construct a correct example.
- Define empirical probability, state the conditions and construct a correct example.
- Define theoretical versus experimental, state the conditions and construct a correct example.
- Define two coins, state the conditions and construct a correct example.
- Define two dice, state the conditions and construct a correct example.
Exercise B — Application and calculations
- Study: A fair coin has probability 1/2 of heads. Verify the calculation or reasoning, then solve a related problem with changed valid values.
- Study: Rolling a fair six-sided die. Verify the calculation or reasoning, then solve a related problem with changed valid values.
- Study: One coin toss: S={H,T}. Verify the calculation or reasoning, then solve a related problem with changed valid values.
- Study: Rolling an even number: E={2,4,6}. Verify the calculation or reasoning, then solve a related problem with changed valid values.
- Study: For a fair die, each face has probability 1/6. Verify the calculation or reasoning, then solve a related problem with changed valid values.
- Study: P(odd on fair die)=3/6=1/2. Verify the calculation or reasoning, then solve a related problem with changed valid values.
- Study: If P(rain)=0.3, P(no rain)=0.7. Verify the calculation or reasoning, then solve a related problem with changed valid values.
- Study: Rolling 7 on a standard six-sided die has probability 0. Verify the calculation or reasoning, then solve a related problem with changed valid values.
- Study: If heads appears 48 times in 100 tosses, experimental probability=0.48. Verify the calculation or reasoning, then solve a related problem with changed valid values.
- Study: 50 tosses need not produce exactly 25 heads. Verify the calculation or reasoning, then solve a related problem with changed valid values.
- Study: P(exactly one head)=2/4=1/2. Verify the calculation or reasoning, then solve a related problem with changed valid values.
- Study: P(sum 7)=6/36=1/6. Verify the calculation or reasoning, then solve a related problem with changed valid values.
Exercise C — Proof, reasoning and challenge
- Prove or justify a central rule using a labelled diagram or clear mathematical argument.
- Give a counterexample to a common false generalisation.
- Compare two related ideas and explain when each applies.
- Write a realistic application with correct units and assumptions.
- Find and correct an error in a hypothetical solution.
- Check an answer using a second valid method.
- Create a three-step question and provide a fully reasoned answer.
- Explain a concept in plain language to a younger student.
Frequently asked questions with answers
What is uncertainty and chance and how is it used?
Probability expresses how likely an event is, from 0 (impossible) to 1 (certain). Example: A fair coin has probability 1/2 of heads.
What is random experiment and how is it used?
A random experiment has known possible outcomes but an uncertain individual result. Example: Rolling a fair six-sided die.
What is sample space and how is it used?
The sample space is the set of all possible outcomes of an experiment. Example: One coin toss: S={H,T}.
What is events and how is it used?
An event is a subset of the sample space. Example: Rolling an even number: E={2,4,6}.
What is equally likely outcomes and how is it used?
The classical formula applies when elementary outcomes are equally likely. Example: For a fair die, each face has probability 1/6.
What is classical probability and how is it used?
For finite equally likely outcomes, P(E)=number favourable/number total. Example: P(odd on fair die)=3/6=1/2.
What is complementary events and how is it used?
P(not E)=1−P(E). Example: If P(rain)=0.3, P(no rain)=0.7.
What is impossible and certain events and how is it used?
An impossible event has probability 0 and a certain event probability 1. Example: Rolling 7 on a standard six-sided die has probability 0.
What is empirical probability and how is it used?
Estimate probability by relative frequency in repeated trials. Example: If heads appears 48 times in 100 tosses, experimental probability=0.48.
What is theoretical versus experimental and how is it used?
Theoretical probability is model-based; experimental estimates vary due to chance and approach stable values over many trials under suitable conditions. Example: 50 tosses need not produce exactly 25 heads.
What is two coins and how is it used?
For two fair independent coins, ordered outcomes are HH, HT, TH, TT. Example: P(exactly one head)=2/4=1/2.
What is two dice and how is it used?
Two independent fair dice have 36 equally likely ordered outcomes. Example: P(sum 7)=6/36=1/6.
What is without replacement and how is it used?
When objects are drawn without replacement, later probabilities depend on earlier draws. Example: Bag 3 red,2 blue: P(red then red)=3/5×2/4=3/10.
What is independent events and how is it used?
For independent events A,B, P(A and B)=P(A)P(B). Example: Two fair coin heads: 1/2×1/2=1/4.
What is mutually exclusive events and how is it used?
Events that cannot happen together have P(A∩B)=0; P(A or B)=P(A)+P(B). Example: On one die, P(1 or 6)=2/6.
Revision checklist
- I can explain and apply uncertainty and chance.
- I can explain and apply random experiment.
- I can explain and apply sample space.
- I can explain and apply events.
- I can explain and apply equally likely outcomes.
- I can explain and apply classical probability.
- I can explain and apply complementary events.
- I can explain and apply impossible and certain events.
- I can explain and apply empirical probability.
- I can explain and apply theoretical versus experimental.
- I can explain and apply two coins.
- I can explain and apply two dice.
- I can explain and apply without replacement.
- I can explain and apply independent events.
- I can explain and apply mutually exclusive events.
- I can explain and apply general addition rule.
- I can explain and apply probability misconceptions.
Official source and edition note
Official NCERT textbook portal. Original supplementary explanations and practice. Exact NCERT exercise-by-exercise answers and any official marking scheme require verification against the prescribed textbook.
