Every Probability Question Is One of a Few Types
Probability looks like a sprawling topic, but GCSE papers recycle a small set of question shapes: work out a single probability, use the fact that probabilities sum to 1, complete a tree diagram or frequency tree, and predict how many times something will happen. Learn to recognise which type you are looking at and the method chooses itself.
All four boards (AQA, Edexcel, OCR and Eduqas) test probability on both tiers. On Foundation it is one of the most reliable places to pick up marks, because the arithmetic is rarely the hard part — the hard part is setting the question up, and that is exactly what this guide covers. If the underlying ideas are new to you, start with our basic probability topic page, which builds the foundations this post moves quickly over.
One convention before we start: a probability is always a number between 0 (impossible) and 1 (certain), written as a fraction, decimal or percentage. An answer of or 1.4 is wrong before the examiner even checks your working, so it is worth a two-second sanity check on every answer.
- Probability questions come in a small number of repeated types — learn the shapes
- Every probability lies between 0 and 1 inclusive
- Fractions, decimals and percentages are all accepted unless the question specifies
- Tested on both tiers by all four exam boards
Single Events: The Basic Formula
For equally likely outcomes, the probability of an event is . The two words that matter are equally likely: the formula works for a fair dice or a well-shuffled pack of cards, and it does not work for a biased spinner, where the question must give you the probabilities instead.
The classic trap in single-event questions is miscounting the favourable outcomes. "A prime number" on a dice means 2, 3 and 5 — students regularly include 1, which is not prime, or forget that 2 is. Listing the outcomes explicitly before counting them costs five seconds and protects the whole question.
Examiners also like wrapping single events in context: a bag of counters, a box of pens, letters of a word. The word MATHEMATICS has 11 letters of which 2 are Ms, so the probability of picking an M at random is . Same formula, different costume.
A fair six-sided dice is rolled once. What is the probability of rolling a prime number?
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List the outcomes: 1, 2, 3, 4, 5, 6 — six equally likely outcomes in total
- 2
Identify the primes: 2, 3 and 5 (remember: 1 is not prime, 2 is)
- 3
Complements: P(not A) = 1 − P(A)
An event either happens or it does not, so the two probabilities must add to 1. That gives the complement rule: . If the probability a train is late is 0.15, the probability it is not late is . This is often a whole question on Foundation papers, worth an easy mark or two.
The rule earns its keep later in the topic, because "not" questions are usually far quicker than the direct route. The phrase to watch for is at least one. Working out the probability of at least one six in several rolls directly means adding up lots of separate cases; working out the probability of no sixes and subtracting from 1 is one calculation. Whenever a question says "at least one", train yourself to think "1 minus the probability of none".
We will use exactly this shortcut in the tree diagram sections below, where it regularly turns a four-branch calculation into a one-branch calculation.
- $P(\text{not } A) = 1 - P(A)$, because the two options cover everything
- "At least one" almost always means: find P(none) and subtract from 1
- Check the complement makes sense — the two probabilities must sum to exactly 1
Mutually Exclusive Events and the Sum-to-1 Question
Events are mutually exclusive when they cannot happen at the same time — a spinner cannot land on red and blue in one spin. For mutually exclusive events, probabilities add: . And if a set of mutually exclusive events covers every possible outcome, their probabilities sum to exactly 1.
That second fact powers one of the most common GCSE question types: a table of probabilities with a missing value. A spinner lands on red, blue or green with and . Since the three probabilities must total 1, . Two marks, done in a line.
Higher papers dress the same idea in algebra: the table might show and alongside known values, and you form and solve an equation by setting the total to 1. The probability content is identical — the sum of the column is 1 — so do not let the intimidate you.
- Mutually exclusive means the events cannot both happen in one trial
- For mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$
- A complete set of mutually exclusive outcomes has probabilities summing to 1
- Table-with-a-missing-value questions: subtract the known probabilities from 1
Tree Diagrams With Replacement
A tree diagram maps out two (or more) events in sequence. Each set of branches shows the possible outcomes of one event, with its probability written on the branch. The two rules that do all the work: multiply along a path of branches to find the probability of that exact sequence, and add between paths when more than one sequence counts as success.
"With replacement" means the first object goes back before the second draw, so the second draw is identical to the first — the events are independent and the probabilities on the second set of branches are the same as the first. This is the easier version, and it is the one to master first. Our tree diagrams topic page generates practice at both difficulties.
A good habit: after completing any tree, check that each set of branches sums to 1, and that the probabilities of all the final outcomes sum to 1. In the example below the four end probabilities are , , and , which total exactly 1 — if yours do not, a branch is wrong.
A bag contains 3 red and 7 blue balls. A ball is taken at random, replaced, then a second ball is taken. Find the probability that the two balls are different colours.
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With replacement, both draws have and
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Multiply along the branches:
- 3
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Add the two successful paths:
Tree Diagrams Without Replacement
"Without replacement" means the first object stays out, so the second draw happens from a smaller pool and the branches change. Both the numerator and the denominator can move: take a red ball from a bag of 4 red and 6 blue, and the second draw is from 9 balls of which only 3 are red. Writing the second-draw probabilities is where the marks are won and lost, so slow down at exactly that step.
The multiply-along, add-between rules are unchanged. In the 4-red, 6-blue bag, , and .
This is also where the complement shortcut from earlier pays off, as in the worked example below: "at least one red" via is a single path instead of three. On the exam, leave fractions unsimplified until the final line — is easier to add to its neighbours than , and examiners accept unsimplified working.
A bag contains 4 red and 6 blue balls. Two balls are taken at random without replacement. Find the probability that at least one ball is red.
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"At least one red" is the complement of "no reds", i.e. both balls blue
- 2
; then 5 blues remain out of 9 balls, so
- 3
- 4
Frequency Trees Are Not Tree Diagrams
Students lose marks by treating these two as the same thing. A tree diagram carries probabilities on its branches — numbers between 0 and 1. A frequency tree carries counts of actual people or objects at its nodes — whole numbers that must add up correctly at every split. If you write inside a frequency tree bubble, or "24 people" on a tree diagram branch, you have mixed the two up.
A typical frequency tree question: 80 students were surveyed; 30 walk to school and the rest do not. Of the walkers, 18 arrive before 8:30. Of the 50 non-walkers, 35 arrive before 8:30. Completing the tree is pure arithmetic: non-walkers, walkers arriving later, non-walkers arriving later. Check: .
The follow-up part usually converts counts back into a probability, and this is where the frequency tree quietly tests conditional thinking. "Given that a student arrived before 8:30, what is the probability they walked?" restricts you to the early arrivers, so the answer is — the denominator is the group named after "given that", not the full 80. Reading the denominator off the wrong group is the single most common error on this question type.
- Tree diagrams hold probabilities on branches; frequency trees hold counts at nodes
- Every split in a frequency tree must add up — check the totals as you go
- "Given that…" questions: the denominator is the size of the given group, not the whole sample
- From the survey above: $P(\text{walked} \mid \text{before 8:30}) = \frac{18}{53}$
Expected Outcomes and Exam Technique
The last recurring type asks you to predict results: expected number of outcomes = probability × number of trials. A biased dice with rolled 300 times is expected to show a six times. The word "expected" is doing careful work here — 45 is the long-run average, not a guarantee, and a one-mark follow-up often asks you to say exactly that.
The same idea runs in reverse as relative frequency: if an experiment produced 45 sixes in 300 rolls, the estimated probability is , and the estimate becomes more reliable as the number of trials grows. Questions comparing experimental results with theoretical probability nearly always want the phrase "more trials gives a better estimate" somewhere in your answer.
Across all these types, the marking pattern is consistent: method marks for correct branch probabilities or a correct product, accuracy marks for the final value. Show the multiplication before evaluating it. If probability keeps costing you marks, it makes a good target for a short, focused revision block — our guide to common GCSE maths mistakes covers the errors examiners report most, and Exam Ladder's practice engine draws probability questions at every difficulty from a bank of over 280,000 generated questions across 70 GCSE topics, so you can drill exactly the question type that trips you up.
- Expected outcomes = probability × number of trials
- Relative frequency estimates probability from experimental results, improving with more trials
- Show the calculation before the answer — the method mark survives an arithmetic slip
- An expected value is a long-run average, not a promise about any single experiment
Frequently asked questions
When do I add probabilities and when do I multiply?
Multiply for AND: the probability that one event happens and then another happens along the same path. Add for OR: when several separate outcomes each count as success, for mutually exclusive events. On a tree diagram this becomes: multiply along the branches, add between the paths.
What changes on a tree diagram without replacement?
The second set of branches uses updated probabilities, because the first object is not returned. Both the numerator and denominator can change: after taking a red from 4 red and 6 blue, the second draw is from 9 balls with only 3 reds left. With replacement, the second branches simply repeat the first.
What is the difference between a frequency tree and a tree diagram?
A frequency tree records counts of actual people or objects at each node, and the numbers must add up at every split. A tree diagram records probabilities between 0 and 1 on each branch. Exam questions often use a completed frequency tree to ask a probability question afterwards.
How do I answer 'at least one' probability questions quickly?
Use the complement: P(at least one) = 1 minus P(none). Working out P(none) is a single path on the tree diagram, which is far quicker than adding every path that contains at least one success. This shortcut works with and without replacement.
Is conditional probability on Foundation or Higher?
Formal conditional probability using P(A|B) = P(A and B) divided by P(B) is Higher tier. Foundation students meet the idea informally through tree diagrams without replacement and 'given that' questions on frequency trees and two-way tables.