Class 9 › Mathematics › Chapter
The Mathematics of Maybe: Introduction to Probability
Randomness, the probability scale, experimental and theoretical probability, sample spaces, events and tree diagrams.
1. What is Probability?
Probability is a measure of how likely an event is to happen, in the same way length or area measures size. It is the area of mathematics that studies randomness.
Probability gives a number to "how sure are we that this will happen?". Is it impossible, unlikely, even, likely or certain?
Random
Randomness
A situation where we know all the possible results but cannot predict which one will happen (tossing a coin, rolling a die, a lucky draw, rain tomorrow).
Random experiment
Trial
An action that can be repeated, whose result may differ each time and cannot be known in advance. Each performance of it is a trial.
Subjective
Personal judgement
A guess based on how one person reads the evidence ("the sky is clear so rain is unlikely"). Different people can give different answers.
Objective
Based on evidence
Measured from recorded data or from reasoning about outcomes. Everyone who uses the same data gets the same answer. This chapter studies objective probability.
- Why a coin toss is fair for deciding who bats first: both outcomes are equally likely and nobody can predict or control the result, so neither team has an advantage.
- Events such as rain are random because they depend on many sensitive factors (temperature, humidity, wind, pressure). We cannot say for sure, but data lets us estimate the chance.
2. The Probability Scale
Probability is measured on a number line from $0$ to $1$.
Range of probability
$$0 \le P(E) \le 1$$
- $P(E) = 0$: the event is impossible (it can never happen).
- $P(E) = 1$: the event is certain (it always happens).
- $P(E) = 0.5$: even chance ($50\%$): as likely to happen as not.
- Close to $0$: less likely (unlikely); close to $1$: more likely.
- To convert to a percentage, multiply by $100$ (for example $0.75 = 75\%$).
A probability can never be negative and can never be more than $1$. An answer such as $-0.2$ or $1.4$ means a mistake.
3. Two Ways to Measure Probability Objectively
| Experimental (statistical) probability | Theoretical probability | |
|---|---|---|
| Based on | Evidence: actual trials or past data | Reasoning about outcomes that are equally likely |
| Needs an experiment? | Yes (or recorded data) | No |
| Formula | $\dfrac{\text{times the event occurred}}{\text{total number of trials}}$ | $\dfrac{\text{favourable outcomes}}{\text{all possible outcomes}}$ |
| Also called | Relative frequency | "Ideal" or classical probability |
4. Experimental Probability
Outcome: a single result of an experiment. Sample space: the set of all possible outcomes (written inside brackets, separated by commas).
Experimental probability (relative frequency)
$$P(\text{event}) = \frac{\text{Number of times the event occurred}}{\text{Total number of trials}}$$
Do the experiment many times and count how often the event happens. The fraction of times it happened is your estimate of its probability.
- Statistical data: the same formula is used with survey or past data (for example, the share of students who prefer a certain item in a sample). Data is used heavily in business, forecasting, insurance and research.
- Population and sample: the population is everyone (or everything) we want to know about; checking all is often impractical, so we collect evidence from a sample. This is called sampling.
- Estimating for the population: expected number in population $\approx P \times \text{population size}$, where $P$ is the probability found from the sample.
- Better estimates: use a larger sample that is representative (not biased towards one group).
Law of Large Numbers
As the number of trials increases, the experimental probability tends to get closer to the theoretical probability.
- With only a few trials, the experimental value can be quite different from the theoretical value, even for a perfectly fair coin or die.
- Example idea: $12$ rolls may not show $\tfrac16$ for each face, but $6000$ rolls will be much closer.
5. Theoretical Probability
Theoretical probability of event $A$
$$P(A) = \frac{\text{Number of favourable outcomes}}{\text{Number of possible outcomes}} = \frac{n(A)}{n(S)}$$
If all outcomes have the same chance, the probability of an event is simply "how many outcomes we want" out of "how many outcomes there are". No experiment is needed.
- Condition: works only when all outcomes are equally likely. Having two outcomes does not make them equally likely (a car "starts" or "does not start" are not equally likely; a marble drawn from a bag with unequal numbers of colours is not equally likely to be each colour, though each marble is).
- Fair / unbiased: a coin or die is fair when it is symmetrical, so no side is favoured. A random toss means it falls freely, without interference.
- Count favourable outcomes carefully; if an object repeats (like the letter B twice in a word), each copy is a separate outcome.
Complement
"Not $A$"
$P(\text{not } A) = 1 - P(A)$. Useful when "not" is easier to count (the event and its opposite together cover all outcomes).
Total
Probabilities add to 1
The probabilities of all the individual outcomes (equally likely or not) add up to $1$.
Impossible / certain
$0$ and $1$
An event with no favourable outcome has $P = 0$; an event containing every outcome has $P = 1$.
Final answer
Format
Give as a fraction in lowest terms (or a decimal / percentage if asked).
6. Independent Trials and the Gambler's Fallacy
A coin or die has no memory. What happened in earlier throws does not change the chance of the next throw.
- Independent events: the result of one trial does not affect the next (repeated coin tosses or die rolls).
- Gambler's fallacy: the wrong belief that after a run of one outcome, the opposite is "due". For a fair coin, $P(\text{tails})$ stays $\tfrac12$ even after six heads in a row; for a fair die $P(6)$ stays $\tfrac16$ after three sixes.
- What probability does tell you: not what will happen next, but what happens in the long run over many trials.
7. Sample Space
Sample space $S$: the set of all possible outcomes of an experiment. Each outcome is an element of $S$. The number of elements is the sample size $n(S)$.
- $S$ must include every possible outcome, and no outcome is listed twice.
- Write it in set brackets: $S = \{\text{outcome}_1, \text{outcome}_2, \dots\}$.
- Level of detail matters: choose a sample space fine enough for the question. If only "rain or no rain" matters, $S = \{\text{Rain}, \text{No Rain}\}$; if the amount matters, expand it (no rain, drizzle, light rain, heavy rain).
| Experiment | Sample space | $n(S)$ |
|---|---|---|
| Toss one coin | $\{H, T\}$ | $2$ |
| Roll one die | $\{1, 2, 3, 4, 5, 6\}$ | $6$ |
| Toss two coins | $\{HH, HT, TH, TT\}$ | $4$ |
| Toss three coins | $\{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$ | $8$ |
| Roll two dice | All pairs $(a, b)$ with $a, b \in \{1, \dots, 6\}$ | $36$ |
Counting rule: for a multi-step experiment, $n(S)$ = (number of outcomes at step 1) $\times$ (number at step 2) $\times \cdots$. For $n$ coin tosses, $n(S) = 2^n$. Order matters: $HT$ and $TH$ are different outcomes. When items are drawn without replacement, the number of choices decreases at each step.
8. Events
Event: any outcome or group of outcomes of a random experiment. In set language, an event $E$ is a subset of the sample space: $E \subseteq S$.
An event is the set of outcomes we are interested in. To find its probability, list those outcomes and count them.
- Describing an event: write the condition in words, then list the outcomes that satisfy it (for example, "at least one head" in two tosses is $\{HH, HT, TH\}$).
- Simple event: an event with exactly one outcome.
- Certain event: $E = S$ (so $P = 1$). Impossible event: $E$ has no elements (so $P = 0$).
- Then $P(E) = \dfrac{n(E)}{n(S)}$ when the outcomes in $S$ are equally likely.
- "At least one" means one or more; "exactly" means that number only; "greater than $n$" does not include $n$.
9. Tree Diagrams
Tree diagram: a diagram that lists all outcomes of a multi-step experiment (a series of trials such as tossing a coin twice or rolling a die three times). Each path from start to end is one complete outcome.
Tree diagram for tossing a fair coin twice: $4$ paths, so $S = \{HH, HT, TH, TT\}$.
How to draw and use a tree diagram
1. From one starting point, draw a branch for each possible outcome of the first step.
2. From the end of each branch, draw branches for every possible outcome of the next step. Repeat for further steps.
3. Read each path from start to end to list the sample space; count paths for $n(S)$.
4. Write the probability of each outcome on its branch. Then $P(E) = \dfrac{\text{paths favourable to } E}{\text{total paths}}$ (when paths are equally likely).
- Multiplying along a path: the probability of a whole path is the product of the probabilities on its branches (for the coins above: $\tfrac12 \times \tfrac12 = \tfrac14$). The probability of an event with several paths is the sum of those path probabilities.
- With replacement (item is put back): the branch probabilities are the same at every step. Without replacement (item is set aside): the second-step probabilities change because there is one item fewer.
- Tree diagrams are also useful for combinations, like choosing one snack and one drink, or shirts and trousers; a table can show the same pairs.
10. Probability Using Area (Dropping an Object on a Region)
When an object lands "at random" on a region, there are infinitely many positions, so we cannot count outcomes. Instead, a part of the region that is bigger is proportionally more likely.
Area model
$$P(\text{lands in a part}) = \frac{\text{area of that part}}{\text{area of the whole region}}$$
- For example, the chance of landing inside a circle drawn within a rectangle equals (area of circle) $\div$ (area of rectangle), using $\pi r^2$ and length $\times$ width.
11. Formula Sheet and Common Mistakes
| Idea | Result |
|---|---|
| Range | $0 \le P(E) \le 1$; impossible $= 0$, certain $= 1$ |
| Experimental probability | $\dfrac{\text{times event occurred}}{\text{total trials}}$ (relative frequency) |
| Theoretical probability | $\dfrac{n(E)}{n(S)}$ (equally likely outcomes) |
| Complement | $P(\text{not } E) = 1 - P(E)$ |
| Estimate for a population | $P \times$ population size |
| Sample space size, multi-step | Multiply the outcomes of each step ($2^n$ for $n$ coins) |
| Event | A subset of $S$ |
| Tree diagram | Each path is an outcome; multiply along a path, add paths of an event |
| Area model | $\dfrac{\text{favourable area}}{\text{total area}}$ |
Common mistakes:
• Giving a probability above $1$ or below $0$.
• Assuming every set of outcomes is equally likely (two outcomes do not always mean $50\%$).
• Listing an incomplete sample space, or listing an outcome twice, or ignoring order ($HT \ne TH$).
• Believing in the gambler's fallacy ("tails is due").
• Expecting experimental and theoretical probability to match exactly after only a few trials.
• Forgetting that without replacement the later probabilities change.
• Not simplifying the fraction, or giving a ratio instead of a fraction.
• Giving a probability above $1$ or below $0$.
• Assuming every set of outcomes is equally likely (two outcomes do not always mean $50\%$).
• Listing an incomplete sample space, or listing an outcome twice, or ignoring order ($HT \ne TH$).
• Believing in the gambler's fallacy ("tails is due").
• Expecting experimental and theoretical probability to match exactly after only a few trials.
• Forgetting that without replacement the later probabilities change.
• Not simplifying the fraction, or giving a ratio instead of a fraction.