Combined events involve finding the probability of two or more events occurring. At Higher tier you use the AND rule (multiply) and OR rule (add) and must be able to work with independent and mutually exclusive events.
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Revising combined events the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Combined events GCSE” on the site for the complete guide.
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Key facts to remember
1AND rule (independent events): P(A and B) = P(A) × P(B).
2OR rule (mutually exclusive events): P(A or B) = P(A) + P(B).
3General OR rule: P(A or B) = P(A) + P(B) − P(A and B).
4Two events are independent if one does not affect the other.
5Two events are mutually exclusive if they cannot both happen at the same time.
6All probabilities in a sample space must sum to 1.
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Formulas
AND rule (independent)
P(A ∩ B) = P(A) × P(B)
OR rule (mutually exclusive)
P(A ∪ B) = P(A) + P(B)
General OR rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
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Worked examples
Example 1
A bag contains 3 red and 7 blue balls. A ball is chosen, replaced, then another is chosen. Find P(both red).
Working
P(red) = 3/10
Since replaced, the events are independent.
P(both red) = P(red) × P(red) = 3/10 × 3/10
= 9/100
Answer9/100 or 0.09
Example 2
P(A) = 0.4, P(B) = 0.3. A and B are mutually exclusive. Find P(A or B).
Working
Mutually exclusive means P(A and B) = 0.
P(A or B) = P(A) + P(B) = 0.4 + 0.3 = 0.7
Answer0.7
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Common mistakes
✗Adding probabilities when you should multiply (AND means multiply).
✗Using the simple OR rule when events are NOT mutually exclusive.
✗Not replacing when the question says "without replacement" — events become dependent.
✗Leaving answers greater than 1 — always check your final probability is between 0 and 1.