Overview
What you’ll learn
- Write probabilities as fractions, decimals or percentages, keep them inside 0–1, and find complements with 1 − P(A).
- Estimate probability from relative frequency and predict expected frequency over a number of trials.
- Build sample space diagrams, Venn diagrams and tree diagrams, multiplying along branches and adding across mutually exclusive cases.
- Handle without-replacement draws where the second denominator drops, and read conditional probabilities from tables and Venn diagrams (Extended).
Syllabus checklist (0580)
Statements are paraphrased and grouped by tier. Core is studied by everyone; Extended content is examined on Papers 2 and 4 only.
Core — every candidate
- 8.1Place events on the probability scale from 0 (impossible) to 1 (certain); calculate the probability of a single event as a fraction, decimal or percentage; use P(not A) = 1 − P(A); read probabilities from tables, graphs and two-set Venn diagrams.
- 8.2Use relative frequency as an estimate of probability and calculate expected frequency; use the language of fair, biased and random.
- 8.3List outcomes with sample space diagrams and Venn diagrams; calculate probabilities of two combined events using tree diagrams with replacement.
Extended — Extended candidates only
- 8.1Use P(A) and P(A′) notation and interpret Venn diagrams with more than two sets. (Extended)
- 8.3Use tree diagrams with and without replacement, and combine events with and-and-or structure, multiplying along branches and adding between them. (Extended)
- 8.4Work out conditional probabilities from Venn diagrams, tree diagrams and two-way tables. (Extended)
In 30 seconds
- 1Probabilities live on a scale from 0 (impossible) to 1 (certain) — anything outside that range is wrong by definition.
- 2P(not A) = 1 − P(A) — the complement shortcut usually beats counting.
- 3Expected frequency = probability × number of trials — it may be a decimal, and that is fine.
- 4AND (both happen) → multiply probabilities along a tree branch; OR (one path or another) → add the path probabilities.
- 5Tree branches always sum to 1; on the second set of branches without replacement, the denominators drop by one (e.g. 4⁄9 then 3⁄8).
- 6“At least one” is almost always fastest as 1 − P(none).
- 7Venn “A or B”: P(A) + P(B) − P(A and B) — subtract the overlap counted twice.
- 8Conditional probability asks “of the given group, how many?”: the denominator is the given group, not the total (Extended).
Notes · 1 of 4
The probability scale and single events
Zero to one, fractions to percentages, and the complement trick.
A probability of 0 means impossible, 1 means certain, and ½ sits halfway. Any value below 0 or above 1 is impossible as an answer — a quick sanity check worth marks. For a single event, count the favourable outcomes over the total outcomes: a bag holding 7 red, 5 blue and 3 green counters (15 in all) gives P(blue) = 5⁄15 = ⅓. The same value may be written as a fraction, a decimal (0.333…) or a percentage (33.⅓%) — the question says which to use.
The complement rule P(not A) = 1 − P(A) turns awkward counts into one subtraction: P(not green) = 1 − 3⁄15 = ⅘. “At least”, “not” and “something else” questions are complements in disguise. On Venn diagrams the same logic shades the outside of the circle for “not A”.
Notes · 2 of 4
Relative and expected frequency
Probability estimated from experiments — and the fair test.
Relative frequency estimates a probability from data: relative frequency = number of successes ÷ number of trials. A die landing six on 31 of 150 rolls estimates P(six) ≈ 31⁄150 ≈ 0.207. More trials → a more trustworthy estimate. If the relative frequency lands close to the theoretical value, the die or spinner behaves fairly; a big, persistent gap suggests bias.
The reverse direction is expected frequency: expected = probability × number of trials. A fair die rolled 90 times should show six about 90 × ⅙ = 15 times. Expected values may be decimals (a spinner with P(win) = 0.25 spun 50 times “expects” 12.5 wins) — do not round to whole numbers unless the question asks.
Notes · 3 of 4
Combined events: sample spaces, Venn diagrams and trees
Multiply along branches, add across paths.
For two independent events, multiply: two coins give P(two heads) = ½ × ½ = ¼. A sample space diagram (a grid of all outcome pairs) makes counting automatic: with two dice, 36 cells, and the diagonal sum of 7 holds 6 of them, so P(sum = 7) = 6⁄36 = ⅙. Venn diagrams handle overlapping sets: P(A or B) = P(A) + P(B) − P(A and B), the subtraction removing the double-counted overlap.
Tree diagrams organise the multiplication. Each pair of branches carries its own probabilities (summing to 1), and each final path is found by multiplying along it; different ways of getting the same description are added. Core papers use replacement (branch probabilities repeat); Extended papers add without replacement, where the second draw’s denominators drop — from a bag of 5 red and 4 blue, P(both blue) = 4⁄9 × 3⁄8 = ⅙, not (4⁄9)².
Notes · 4 of 4
Conditional probability
“Given that…” shrinks the denominator. (Extended)
Extended. A conditional question changes the denominator: “Given that a student wears glasses, what is the probability they are a girl?” counts only inside the glasses group. No formula is needed — read it off a two-way table or a Venn diagram. If 12 students wear glasses and 7 of them are girls, the answer is 7⁄12, not 7⁄30: the glasses-wearers are the whole new sample space.
On tree diagrams, conditional probability is the second-stage branch given the first: P(¬red second | red first) is just the branch label 5⁄8 after a red first draw. On Venn diagrams, restrict attention to the given set and take the ratio inside it. The only repeated trap is reverting to the overall total — underline the “given” phrase and let it choose the denominator.
Revise
Key terms
The words examiners expect you to use precisely.
- Probability scale
- The number line from 0 (impossible) to 1 (certain).P(raining indoors) = 0.
- Complement
- The event “A does not happen”: P(A′) = 1 − P(A).P(not red) = 1 − 7⁄15.
- Relative frequency
- Successes ÷ trials, used to estimate probability.31 sixes in 150 rolls → 0.207.
- Expected frequency
- Probability × number of trials.0.25 × 60 spins = 15.
- Sample space
- The list or grid of all possible outcomes.Two dice: 36 equally likely cells.
- Mutually exclusive
- Events that cannot happen together; their probabilities add.A die shows a 3 or a 4, never both.
Show 2 more key termsShow fewer key terms
- Independent
- One event does not change the other’s probability.Coin and die: multiply freely.
- Conditional probability (Extended)
- Probability within a reduced sample space: “given that…”.28 of the 38 finishers → 14⁄19.
Revise
Common mistakes
Where students most often lose marks in this topic — and what to do instead.
Mistake: Giving a probability above 1 or below 0.
The scale only runs from impossible (0) to certain (1).
Instead: Any answer outside 0–1 signals a counting error — recheck.
Mistake: Adding probabilities when events must both happen.
“A and B” multiplies along the branches; “A or B” adds distinct paths.
Instead: AND → multiply; OR (mutually exclusive) → add.
Show 10 more mistakesShow fewer mistakes
Mistake: Keeping the same denominator on the second draw without replacement.
One counter has gone, so 9 counters become 8.
Instead: Drop both numerator and denominator: 4⁄9 then 3⁄8.
Mistake: Adding the overlap twice in a Venn “or” question.
P(A) + P(B) counts the intersection two times.
Instead: P(A or B) = P(A) + P(B) − P(A and B).
Mistake: Grinding out every “at least one” path instead of using the complement.
Long paths invite arithmetic slips and cost time.
Instead: P(at least one) = 1 − P(none).
Mistake: Using the theoretical probability when the question says “estimate”.
“Estimate” asks for the relative frequency from the trial data.
Instead: Estimate = successes ÷ trials; use it for later predictions.
Mistake: Rounding an expected frequency to a whole number automatically.
Expected values describe long-run averages and may be decimals.
Instead: 0.25 × 50 = 12.5 is a fine answer unless told otherwise.
Mistake: Answering a conditional question with the overall total in the denominator.
“Given that…” shrinks the sample space to the given group.
Instead: Read the row or circle of the given group first; that total is the new denominator.
Mistake: Writing tree branches whose probabilities do not sum to 1.
Each split must cover every possibility exactly.
Instead: Win 0.6 pairs with lose 0.4; check each fork before multiplying.
Mistake: Multiplying probabilities for events connected by “or”.
Multiplication is for “and”; “or” adds (after subtracting any overlap).
Instead: Mutually exclusive: P(A or B) = P(A) + P(B); 0.3 + 0.45 = 0.75, not 0.135.
Mistake: Keeping the full denominator in a conditional probability.
The condition shrinks the sample space before the second stage.
Instead: Given the first counter is black from 4 black and 6 white, P(second black) = 3⁄9, not 3⁄10.
Mistake: Dividing successes by failures when estimating a probability.
Relative frequency compares successes with all attempts.
Instead: 31 sixes in 150 rolls estimate 31⁄150 ≈ 0.207, not 31⁄119.
Check yourself
Quick quiz
24 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Place impossible events at 0 on the probability scale
The probability of an impossible event is
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Practise
Topical questions
14 exam-style questions with model answers and mark schemes.
A set of structured and calculation questions matched to this topic, in three tiers, each with a model answer, mark scheme and examiner tip.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a Cambridge paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Topic 17 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Topic 17: the probability scale and complements, relative and expected frequency, sample space diagrams, Venn diagrams, tree diagrams with and without replacement, at-least-one complements, and conditional probability from two-way tables.
Mark scheme
PROTopic 17 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.
Which papers examine this topic?
Paper 1 / 3
Core: 80 marks each, 1 h 30 each
Paper 1 non-calculator, Paper 3 calculator
Paper 2 / 4
Extended: 100 marks each, 2 h each
Paper 2 non-calculator, Paper 4 calculator
Accuracy
3 s.f., angles to 1 d.p.
Unless a question states otherwise
Want a full paper? Try the Paper 1-style Core, Paper 2-style non-calculator or Paper 4-style calculator sets.
Questions
Frequently asked questions
Should I write probabilities as fractions, decimals or percentages?
Use whatever the question asks for; when it does not, fractions are usually safest because they stay exact. A probability of ⅓ becomes 0.333… or 33⅓% — rounding too early (0.33) can lose the accuracy mark later, so keep fractions through the working and convert at the end.
When do I multiply and when do I add?
Multiply when the events must **both** happen along one path (A **and** B); add when several **distinct paths** count towards the answer (this path **or** that path). On a tree diagram: multiply along the branches, add between the finished paths. “At least one” is usually fastest as 1 − P(none).
What exactly changes when items are drawn without replacement?
The second stage’s probabilities: one item has gone, so every denominator drops by one (4⁄9 becomes 3⁄8), and if that colour was taken its numerator drops too. With replacement the branches are identical — that is the whole difference between the two tree diagrams.
Is relative frequency better than the theoretical probability?
They answer different questions. The theoretical probability is the ideal long-run value when outcomes are equally likely; relative frequency is evidence from real trials and is the only option when you cannot compute the theory (a biased die, a medical trial). The more trials, the closer the estimate should sit to the true value — a small gap is normal, a persistent large gap suggests bias.
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