Probability measures how likely an event is, from 0 for impossible to 1 for certain. For equally likely outcomes, count the successful outcomes and divide by the total number of outcomes. Expected frequency describes a long-run average, so an actual experiment can give a different result. When two events occur together, multiply their probabilities if the events are independent. Independence means one result does not change the probability of the other. For mutually exclusive alternatives, add probabilities. List outcomes or draw a tree first if you are unsure what the events include.
Learning objectives
- Calculate probabilities and use complementary outcomes.
- Find an expected frequency and combine independent events.
Key ideas
For equally likely outcomes, count favourable outcomes out of the total.
Probabilities lie between zero and one; all mutually exclusive exhaustive outcomes total one.
The probability of an event not happening is its complement.
An expected frequency predicts a long-run average, not a guaranteed result.
Multiply probabilities to combine independent events; without replacement, probabilities may change.
Worked examples
Example 1
Predict the number of even results in ninety rolls of a fair six-sided die.
Answer
Example 2
A fair coin and fair six-sided die are independent. Find the probability of heads and a six.
Answer
Question styles to prepare for
Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.
AO1 · Methods
Calculate a probability or expected frequency using a sample space or table.
AO2 · Reasoning
Explain whether outcomes are equally likely or a probability model is suitable.
AO3 · Problem solving
Use a probability model to estimate a count and discuss its limitations.
When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.
Avoid this mistake
Expected frequency is not a promise. Multiplication of unchanged probabilities needs independent events.
Try it yourself
Attempt each question before opening its answer.
Check 1: P(rain tomorrow) = 0.3. What is P(no rain tomorrow)?
Answer: 0.7
Rain and no rain cover all possibilities, so subtract 0.3 from 1.
Check 2: A fair coin and a fair six-sided die are used independently. Find P(heads and a 6).
Answer: 1/12
P(heads) = 1/2 and P(6) = 1/6. Multiply: 1/2 × 1/6 = 1/12.
Explain it in your own words
- Describe an impossible event, a certain event and an event with probability 1/2.
- Explain why an expected frequency need not equal the observed frequency.
- Give one example of independent events and one where a first draw changes the next probability.
Keep practising
Download the compact topic guide (PDF) · Practise this topic in Blitz · All foundation notes
Specification source
AQA Probability specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.