Overview
What you’ll learn
- Use the language of experiments, outcomes, sample spaces and events correctly.
- List the sample space of a simple experiment and count favourable outcomes.
- Express the probability of a single event as a fraction, decimal or percentage.
- Use the complement rule P(not A) = 1 − P(A).
- Estimate probabilities from experiments using relative frequency.
- Calculate expected outcomes as probability × number of trials, and comment on results.
In 30 seconds
- 1The sample space lists every possible outcome; an event is the collection of favourable ones.
- 2For equally likely outcomes: P(event) = favourable outcomes ÷ total outcomes.
- 3Probabilities always live on a scale from 0 (impossible) to 1 (certain) — ½ and 50% are the same even chance.
- 4The complement shortcut: P(not A) = 1 − P(A) — often the faster count.
- 5Experimental probability = successful trials ÷ total trials; it settles towards the true probability as trials pile up.
- 6Expected outcomes = probability × number of trials — an average prediction, not a guarantee.
Notes · 1 of 6
The language of chance
Experiments produce outcomes; the full list is the sample space.
| Term | Meaning | Example (one die) |
|---|---|---|
| Experiment | a repeatable random action | rolling the die once |
| Outcome | one possible result | showing a 4 |
| Sample space | the list of all outcomes | {1, 2, 3, 4, 5, 6} |
| Event | the outcomes we care about | rolling an even number |
Probability is measured on a scale from 0 to 1: 0 means impossible, 1 means certain, and ½ (equivalently 0.5 or 50%) is an even chance. Words like “likely” get numbers; the number is the precision.
Notes · 2 of 6
Theoretical probability
Count favourable outcomes, divide by total outcomes.
- 11. Write the sample space
List every equally likely outcome — no duplicates.
- 22. Count the total
How many outcomes are there altogether?
- 33. Count favourable
How many of them match the event?
- 44. Divide
P(event) = favourable ÷ total, then simplify.
| Event | Favourable | P(event) |
|---|---|---|
| Rolling a 3 | {3} → 1 | 1/6 |
| Rolling an even number | {2, 4, 6} → 3 | 3/6 = 1/2 |
| Rolling a number under 3 | {1, 2} → 2 | 2/6 = 1/3 |
Notes · 3 of 6
Fractions, decimals and percentages
Three costumes for the same number — pick the costume the question asks for.
| Event | Fraction | Decimal | Percentage |
|---|---|---|---|
| Rolling an even number | 1/2 | 0.5 | 50% |
| Drawing a red sweet (3 of 10) | 3/10 | 0.3 | 30% |
| Rolling a 6 | 1/6 | 0.167 (2 d.p.) | 16.7% (1 d.p.) |
A quick self-check for any answer: every probability must satisfy 0 ≤ P ≤ 1. A value like 1.2 or −0.3 is instantly wrong — probabilities are shares of one whole, so they cannot leak outside [0, 1].
Notes · 4 of 6
The complement
Not happening is the one whole minus happening.
The complement of an event is “the event does not happen”. Because something must happen, the two probabilities split the whole: P(not A) = 1 − P(A). When “not A” is easier to count — often far easier — count it and subtract.
| Event A | P(A) | P(not A) |
|---|---|---|
| Rolling a 6 | 1/6 | 5/6 |
| Rolling an even number | 1/2 | 1/2 |
| Rolling a number under 7 | 1 | 0 |
Notes · 5 of 6
Experimental probability
When theory is unavailable, let the trials talk.
| Landing | Count | Relative frequency |
|---|---|---|
| Point up | 140 | 140 ÷ 250 = 0.56 |
| Point down | 110 | 110 ÷ 250 = 0.44 |
Experimental probability (relative frequency) = successful trials ÷ total trials. A drawing pin is not symmetric, so no fair-share formula exists — the only honest estimate is the one the experiment gives. With more trials the estimate settles down towards the true probability; short runs wobble.
Notes · 6 of 6
Expected outcomes
Multiply the probability by the number of trials to predict the long-run count.
The expected number of successes = probability × number of trials. It is an average prediction: spinning a winner with P = 0.2 two hundred times expects 0.2 × 200 = 40 wins — not a promise of exactly 40 on the day.
| Situation | Probability | Trials | Expected |
|---|---|---|---|
| Sixes on a die | 1/6 | 300 rolls | 50 sixes |
| Rain on a wet-season day | 0.3 | 30 days | 9 days |
| Free throw at 80% | 0.8 | 15 shots | 12 shots |
Revise
Key terms
The words examiners expect you to use precisely.
- Experiment
- A repeatable action with an unpredictable result, such as rolling a die.
- Outcome
- One possible result of an experiment.
- Sample space
- The complete list of all possible outcomes.
- Event
- A chosen set of favourable outcomes.
- Probability
- A number from 0 to 1 measuring how likely an event is.
- Equally likely
- Outcomes with identical chance; the licence for the counting formula.
Show 6 more key termsShow fewer key terms
- Fair
- A die, coin or spinner whose outcomes are equally likely.
- Complement
- The event “A does not happen”; P(not A) = 1 − P(A).
- Relative frequency
- Successful trials ÷ total trials; the experimental estimate of a probability.
- Experimental probability
- A probability estimated from trials rather than from symmetric counting.
- Theoretical probability
- A probability computed by counting equally likely outcomes.
- Expected number
- Probability × number of trials — the long-run average count of successes.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Writing probabilities greater than 1 or below 0.
A probability is a share of one whole; it lives between 0 and 1.
Instead: Check 0 ≤ P ≤ 1 every time; “7 out of 5” is a sign the totals were mixed up.
Mistake: Dividing favourable outcomes by the wrong total (e.g. by the remaining outcomes).
The formula divides by the whole sample space.
Instead: P(event) = favourable ÷ total outcomes of the full experiment.
Show 8 more mistakesShow fewer mistakes
Mistake: Listing a sample space with duplicates, like heads–heads and heads–tails mixed up for two coins.
Unequal lists make every probability that follows wrong.
Instead: List systematically and count each distinct outcome once; two coins have 4 outcomes, not 3.
Mistake: Counting outcomes for “not A” by hand when 1 − P(A) is available.
The long way invites slips, especially when “not A” has many outcomes.
Instead: P(not A) = 1 − P(A): compute the easy side, subtract from 1.
Mistake: Assuming an experiment proves bias after a few unusual trials.
Short runs wobble far from the true probability by pure chance.
Instead: Compare the relative frequency with the theoretical value, then gather many more trials before judging.
Mistake: Treating an expected value as a guarantee (“20 wins must happen”).
Expectation is an average over many repeats, not a promise for one session.
Instead: Say “expected about 20”; real results cluster near it but rarely land exactly on it.
Mistake: Giving a probability answer without the requested form.
A fraction answer loses the mark when the question says “as a percentage”.
Instead: Convert to the costume the question names: fraction, decimal, or percentage.
Mistake: Expecting a six “because it has not come up for five rolls”.
The die has no memory — each roll is independent, and the chance stays 1/6 however long the drought.
Instead: Independent trials: past results never change the next probability.
Mistake: Reporting P(even) on a die as 3 “because three numbers are even”.
A probability is a share of the whole sample space, not a raw count.
Instead: Divide favourable by total: 3 ÷ 6 = 1/2.
Mistake: Demanding exactly 5 heads from 10 fair flips.
Expectation is an average over many repetitions, not a contract for one short session — 4, 6 and 7 heads are all common.
Instead: Expect about 5; results scatter around it, settling closer only as trials pile up.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Read the probability scale
A probability of 0 means the event is
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 5Foundation
- 6Standard
- 4Challenging
15 questions worth 59 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 11 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 11.
Mark scheme
PROChapter 11 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 3Coming soonWeighted Assessment 3 (WA3)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 2 Math Chapter 11 about?
Chapter 11 covers the probability of single events: sample spaces, the equally-likely counting formula, probabilities as fractions, decimals and percentages, the complement rule, and experimental probability with expected outcomes.
What is the difference between theoretical and experimental probability?
Theoretical probability comes from counting equally likely outcomes (a fair die gives P(6) = 1/6). Experimental probability comes from actually trying it — successful trials divided by total trials. With enough trials, the experimental value settles near the theoretical one.
How do I find the probability of an event NOT happening?
Subtract from 1: P(not A) = 1 − P(A). If P(rain) = 0.3, then P(no rain) = 0.7. This complement shortcut is usually faster than counting all the “not” outcomes.
If a probability says 40 wins are “expected”, must exactly 40 happen?
No. Expected number = probability × trials is a long-run average. Real sessions land near it — sometimes above, sometimes below — and only a very long run shows how accurate the prediction was.