GCSE Maths Probability Notes
Probability measures how likely an event is, from 0 (impossible) to 1 (certain). Careful counting of outcomes is the key to reliable marks.
Key concepts
- Probability scale
- All probabilities lie between 0 and 1, written as fractions, decimals or percentages.
- Total probability
- The probabilities of all possible outcomes add up to 1.
- Expected frequency
- Expected number = probability × number of trials.
- Mutually exclusive events
- Events that cannot happen together — add their probabilities for 'or'.
- P(not A) = 1 − P(A).
- For independent events, multiply probabilities for 'and'.
- Relative frequency estimates probability from experimental results.
Key formulas
Probability of an event
P(event) = favourable outcomes ÷ total outcomes
Expected frequency
expected = P(event) × trials
Worked example
A bag has 3 red and 5 blue counters. One is taken at random. Find P(red).
- 1Count favourable outcomes: 3 red.
- 2Count total outcomes: 3 + 5 = 8.
- 3Write the probability: 3/8.
Answer: 3/8
Common mistakes
✗ Probabilities that do not add to 1.
✓ Check all outcomes sum to 1; use P(not A) = 1 − P(A).
✗ Adding when you should multiply (or vice versa).
✓ 'And' (independent) → multiply; 'or' (mutually exclusive) → add.
✗ Writing probability greater than 1.
✓ A probability can never exceed 1 — recheck your working.
Quick check
Try these, then reveal the answers.
If P(win) = 0.3, what is P(not win)?Reveal answer
0.7
A fair die: P(rolling a 6)?Reveal answer
1/6
Expected sixes in 60 rolls?Reveal answer
10
Exam tips
- Write probabilities as fractions to keep 'and/or' calculations exact.
- List outcomes systematically so you do not miss any.
- Use a tree diagram for two-stage events and multiply along the branches.
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Frequently asked questions
How do I find the probability of 'A or B'?+
For mutually exclusive events, add the probabilities: P(A) + P(B).
What is expected frequency?+
The number of times you expect an event to happen: probability × number of trials.
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