GCSE Maths • Foundation & Higher
GCSE Probability Revision Guide
Understand likelihood, theoretical probability and experimental estimates.
Foundation
Theoretical Probability
Experimental Probability
What You'll Learn
Interpret the Probability Scale
Place and compare probabilities from 0 to 1 and use impossible, unlikely, even chance, likely and certain accurately.
Calculate Basic Probability
Count favourable and total outcomes when all possible outcomes are equally likely.
Use Complements and Totals
Use P(not A) = 1 - P(A) and the fact that exhaustive mutually exclusive probabilities sum to 1.
Work with Different Forms
Convert probabilities between fractions, decimals and percentages before comparing them.
Calculate Relative Frequency
Use observed frequency divided by the number of trials as an experimental estimate.
Make Predictions
Multiply a probability or relative frequency by a future number of trials and interpret the result as an estimate.
KEY RULES AND FORMULAE Rules
Probability Lessons
Basic Probability
Calculate probabilities from equally likely outcomes and express answers in different forms.
Probability Scale
Interpret and position probabilities between impossible and certain.
Relative Frequency
Estimate probabilities from experiments, compare samples and predict future frequencies.
CORE SUBTOPICS
Events and Outcomes
An outcome is one possible result; an event is a set of one or more outcomes.
Fairness and Equally Likely Outcomes
A fair coin, die or equal-section spinner gives equal chances to its possible outcomes.
Mutually Exclusive Events
Events are mutually exclusive when they cannot happen together in the same trial.
Theoretical vs Experimental Probability
Theoretical probability comes from a model; relative frequency comes from observed data.
Sample Size and Variation
Repeated experiments can produce different results; larger samples generally give more stable estimates.
Systematic Outcome Listing
Tables and sample spaces help ensure no possible outcome is omitted or counted twice.
Worked Example
Experimental Probability and Prediction
QUESTION
A biased spinner is spun 240 times and lands on blue 78 times. (a) Estimate P(blue). (b) Estimate how many blue results would occur in the next 600 spins. (c) Estimate P(not blue).
- Given information: The spinner is described as biased, so use the experimental results rather than assuming equal sections.
- Method: Calculate relative frequency, multiply it by the future number of trials and use the complement rule.
- Relative frequency of blue = 78/240 = 13/40 = 0.325.
- Estimated blue results in 600 spins = 0.325 × 600 = 195.
- P(not blue) = 1 - 0.325 = 0.675.
- Check: 0.325 + 0.675 = 1.
- Final answP(blue) ≈ 0.325; estimated blue results = 195; P(not blue) ≈ 0.675.er: The final price is £748, representing an overall decrease of 6.5%.
- The predicted proportion is 195/600 = 0.325, matching the observed relative frequency.
- Do not calculate 240/78. Relative frequency is event occurrences divided by total trials.
- Common GCSE Mistakes
- Using the Formula Without Equal Likelihood
- Reversing the Relative-Frequency Division
- Giving a Probability Outside 0 to 1
- Confusing Unlikely with Impossible
- Assuming Short Experiments Must Match Theory
- Believing Previous Independent Results Change the Next Trial
- Adding Overlapping Events Directly
- Exam Tips
- Write the event and sample space before counting favourable outcomes.
- Check that every final probability lies between 0 and 1 and use a common form when comparing values.
- Look for words such as fair, equally likely, biased, experimental and relative frequency because they determine the method.
- Use the complement rule when “not”, “does not” or “at least one” can simplify the calculation.
- Treat a predicted frequency as an estimate and keep sufficient accuracy until the final step.
- For experimental data, compare both relative frequencies and sample sizes rather than raw frequencies alone.
Ready to Practice?
Printable Worksheet
Practise probability scales, equally likely outcomes, complements, relative frequency and predictions.
Interactive Quiz
Check probability vocabulary, conversions and common misconceptions with instant feedback.
Exam-Style Questions
Apply probability rules to tables, experiments and multi-step contexts.
Watch the Probability Tutorial
Clear, step-by-step explanation of expanding brackets, simplifying complex expressions, and solving for x. Perfect for visual learners who need a breakdown of the rules in action.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Continue Your GCSE Maths Revision
Continue Your GCSE Maths Revision
Frequently Asked Questions
A probability must be between 0 and 1 inclusive. It may also be written as an equivalent fraction, decimal or percentage.
Use this rule when the complete possible outcomes are known and equally likely. It is not valid merely because outcomes can be listed.
Relative frequency is event occurrences divided by total trials. It estimates probability from observed results rather than a theoretical model.
Probabilities add to 1 when the outcomes are mutually exclusive and exhaustive: exactly one must occur and no possibilities are missing.
No. A larger number of trials generally makes relative frequency more stable and often closer to theory, but random variation remains.
Multiply the probability or relative frequency by the planned number of trials. State that the result is an estimate.
Not for independent tosses of a fair coin. Each new toss still has probability 1/2 of tails; previous outcomes do not make tails due.
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