CLASS 9 • MATHS • CHAPTER 7
The Mathematics of Maybe: Introduction to Probability
Short, colourful, handwritten-style notes by EduGrown
1What is Probability?
📝 Probability
Probability is a type of measurement — just like length, area or volume. But instead of measuring physical things, it measures the likelihood of an event, i.e. how confident we are that it will happen.
✏️ Everyday questions
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- Is it going to rain today?
- Will our school win tomorrow’s hockey match?
- Will my name come up in the lucky draw?
2Randomness & Random Experiments
📝 Randomness
Randomness is a situation where you cannot predict exactly what will happen, even though you know all the possible outcomes.
⭐ Classic examples
- Tossing a coin → it will be Heads or Tails, but which one? Unknown.
- Rolling a die → it will be 1, 2, 3, 4, 5 or 6, but which one? Unknown.
📝 Random experiment (or trial)
An action you can repeat (like tossing a coin), where every time the result may be different and the outcome cannot be known in advance.
💡 Why is rain ‘random’?
Rain depends on so many factors — temperature, humidity, wind, pressure — and is so sensitive to them that exact prediction is impossible. We can still estimate its likelihood from past data.
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3Subjective vs Objective Probability
| Subjective probability | Objective probability |
|---|---|
| Based on personal opinion or how a person reads the situation | Based on evidence — data, experiments or fair reasoning |
| “Sun is bright, so it won’t rain today.” | “It rained on 12 of the last 30 days, so P ≈ 0.4” |
| Can differ from person to person | Same for everyone who uses the same evidence |
💡 Focus of this chapter
We learn to measure probability objectively — by collecting evidence instead of guessing.
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4The Probability Scale (0 to 1)
0Impossible
0.25Less likely
0.5Even chance
0.75More likely
1Certain
⭐ The golden rule
0 ≤ P(E) ≤ 1
- P(E) = 0 → the event is impossible
- P(E) = 1 → the event is certain
- P(E) = 0.5 → equally likely to happen or not
| Event | What it means |
|---|---|
| Getting a number greater than 6 on a die | Impossible — a die has only 1 to 6 |
| Rolling a 3 on a die | Less likely — but not impossible |
| Flipping a coin and getting heads | Even chance — H and T equally likely |
| Drawing a number card 2–10 from 52 cards | More likely — 36 of the 52 cards qualify |
| Choosing a red sweet from a bag of all red sweets | Certain — every sweet is red |
5Two Ways of Measuring Probability
⭐ The two routes
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- Evidence from experience → do the experiment many times, or study past data, and find the relative frequency. This is experimental probability.
- Theoretical method → assume all outcomes are equally likely and simply reason it out. This is theoretical probability. No experiment needed!
6Experimental Probability
⭐ Formula
Experimental Probability =
Number of times the event occurredTotal number of trials
This value is also called the relative frequency of the event.
✏️ Example
A die is rolled 50 times and a 4 shows up 8 times.
Experimental probability of a 4 = 850 = 0.16 or 16%.
Experimental probability of a 4 = 850 = 0.16 or 16%.
💡 When is it the only option?
For a paper cup we have no idea what the fair chances are — the three positions are not equally likely. So theory cannot help; we must toss it many times and count. That is experimental probability.
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7Theoretical Probability
📝 Theoretical probability
It assumes a perfectly fair situation where all outcomes are equally likely. It is written as P(E) or P(Event). No data or experiment is required.
⭐ Formula
P(E) = Number of favourable outcomesNumber of possible outcomes
✏️ Example 1 — rolling a die
P(getting a 4) = 16 = 0.1666… ≈ 0.167 or 16.7%
(1 favourable outcome out of 6 possible outcomes)
(1 favourable outcome out of 6 possible outcomes)
✏️ Example 2 — the word PROBABILITY
A letter is picked at random from PROBABILITY.
Number of B’s = 2, total letters = 11
P(letter B) = 211 = 0.1818… ≈ 0.182 or 18.2%
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Number of B’s = 2, total letters = 11
P(letter B) = 211 = 0.1818… ≈ 0.182 or 18.2%
8Statistical Data, Sample & Population
✏️ The favourite-fruit survey
Out of 50 students: 20 like mango, 15 apples, 10 bananas, 5 grapes.
P(a random student likes mango) = 2050 = 0.4 = 40%
P(a random student likes mango) = 2050 = 0.4 = 40%
📝 Population and Sample
- Population — the whole group we care about (all 1500 students of the school).
- Sample — the smaller group we actually collect data from (the 50 students).
- Choosing and studying such a group is called sampling.
💡 Using a sample to predict
If 40% of the sample like mango, then for the whole school of 1500 students we expect about
40% of 1500 = 600 mangoes to be needed.
🔸 A bigger and more representative sample (students from different classes) gives a more reliable estimate.
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🔸 A bigger and more representative sample (students from different classes) gives a more reliable estimate.
9Law of Large Numbers
⭐ The key idea
Even in a perfectly fair situation, experimental probability can differ from theoretical probability — especially when the number of trials is small.
But as the number of trials increases, the experimental probability gets closer and closer to the theoretical one.
This is called the Law of Large Numbers.
But as the number of trials increases, the experimental probability gets closer and closer to the theoretical one.
This is called the Law of Large Numbers.
| Experimental | Theoretical | |
|---|---|---|
| Based on | actual data from trials | fair, equally likely outcomes |
| Needs an experiment? | Yes | No |
| Changes each time? | Yes, it can | No, always fixed |
10Gambler’s Fallacy
⚠️ The famous mistake
Many people think that if something random happens many times in a row, the opposite must be “due” next. This wrong belief is called the Gambler’s Fallacy.
🔸 A coin comes up heads six times — the chance of tails on the next flip is still 12.
🔸 You roll three 6’s in a row — the chance of a 6 next is still 16 ≈ 0.166.
🔸 A coin comes up heads six times — the chance of tails on the next flip is still 12.
🔸 You roll three 6’s in a row — the chance of a 6 next is still 16 ≈ 0.166.
💡 Remember this line
A coin and a die have no memory. Each toss or roll is an independent event — the past does not change the future.
📜 Did you know?
Snakes and Ladders comes from ancient India! It grew out of a dice game called Jñān-Chaupaḍ, used as a teaching tool — each ladder stood for a virtue and each snake for a vice.
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11Fair and Unbiased
📝 What do these words mean?
- Fair — the coin (or die) is symmetrical, so there is no reason for one side to come up more often.
- Unbiased — the same property, stated as ‘no side is favoured’.
- Random toss — the coin is allowed to fall freely, with no interference.
💡 Why the cricket toss is fair
Because a fair coin gives each captain exactly 50% chance — nobody can control or predict it.
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12Sample Space
📝 Sample space (S)
The set of all possible outcomes of a random experiment. Each outcome is an element of S.
The number of elements is the sample size, written as n(S).
The number of elements is the sample size, written as n(S).
⚠️ Two strict rules
- S must include every possible outcome.
- No outcome may be listed more than once.
| Experiment | Sample space S | n(S) |
|---|---|---|
| Will it rain tomorrow? | {Rain, No Rain} | 2 |
| Result of a match | {Win, Lose, Draw} | 3 |
| Tossing one coin | {H, T} | 2 |
| Rolling one die | {1, 2, 3, 4, 5, 6} | 6 |
| Tossing two coins | {HH, HT, TH, TT} | 4 |
✔ Listing two coins carefully
| Coin 1 | Coin 2 | Outcome |
|---|---|---|
| H | H | HH |
| H | T | HT |
| T | H | TH |
| T | T | TT |
💡 Make S detailed enough
{Rain, No Rain} is fine if we only care whether it rains. But if the amount matters, use
{No Rain, Drizzle, Light Rain, Heavy Rain}. The sample space must match the question being asked.
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13Events
📝 Event (E)
An event is any single outcome, or a group of outcomes, that might happen in a random experiment.
In short: an event is a subset of the sample space.
In short: an event is a subset of the sample space.
| Experiment | Sample space S | Event E |
|---|---|---|
| Tossing two coins | {HH, HT, TH, TT} | ‘At least one Head’ → {HH, HT, TH} |
| Rolling a die | {1, 2, 3, 4, 5, 6} | ‘Number > 4’ → {5, 6} |
| Picking a fruit | {Apple, Banana, Orange} | ‘Fruit is yellow’ → {Banana} |
14Tree Diagrams
📝 Tree diagram
A picture used to list all possible outcomes of a multi-step experiment (a series of independent trials, e.g. tossing a coin twice, rolling a die three times).
Each branch is one outcome; each full path from start to end is one complete result.
Each branch is one outcome; each full path from start to end is one complete result.
⭐ How to draw it
- From one starting point, draw a branch for each outcome of step 1.
- From every branch tip, draw branches for each outcome of step 2.
- Write the probability on each branch.
- Read the outcomes at the ends → that is your sample space.
✏️ Coin tossed twice
S = {HH, HT, TH, TT}, so n(S) = 4.
P(HH) = 14 = 0.25 or 25%
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P(HH) = 14 = 0.25 or 25%
15All Formulas at a Glance
⭐ Keep these on your fingertips
0 ≤ P(E) ≤ 1
Experimental P =
Times the event occurredTotal number of trials
Theoretical P(E) =
Favourable outcomesTotal possible outcomes
| Term | Meaning in one line |
|---|---|
| Outcome | A single result of an experiment |
| Sample space S | Set of all possible outcomes |
| Sample size n(S) | How many outcomes are in S |
| Event E | A subset of the sample space |
| Relative frequency | Experimental probability from data |
| Equally likely | Every outcome has the same chance |
| Population / Sample | Whole group / smaller group studied |
16Common Mistakes to Avoid
❌ Don’t do this
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- Writing a probability greater than 1 or negative. Always 0 ≤ P(E) ≤ 1.
- Falling for the Gambler’s Fallacy — past results never change the next result.
- Repeating an outcome or missing one while writing the sample space.
- Forgetting that HT and TH are different outcomes when two coins are tossed.
- Using the theoretical formula when outcomes are not equally likely (e.g. the paper cup).
- Confusing favourable outcomes with total outcomes in the fraction.
- Expecting experimental probability to exactly match theory after only a few trials.
- Making a sample space too rough for the question being asked.
17Chapter Summary — Quick Revision
🏆 All of Chapter 7 in 10 lines
- Probability measures the likelihood of an event, just as length measures distance.
- An event is random when we know the possible outcomes but not which one will occur.
- Probability lies on a scale from 0 (impossible) to 1 (certain): 0 ≤ P(E) ≤ 1.
- Experimental probability = times the event occurredtotal trials — based on real data.
- Theoretical probability = favourable outcomespossible outcomes — assumes all outcomes are equally likely.
- Statistical probability uses a sample to estimate results for the whole population.
- By the Law of Large Numbers, more trials bring experimental probability closer to theoretical.
- The Gambler’s Fallacy is the wrong belief that past results affect the next one — coins have no memory.
- Sample space S lists every possible outcome once; an event E is a subset of S.
- Tree diagrams list all outcomes of multi-step experiments and help find their probabilities.
✨ Happy Learning — @edugrown ✨