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Mutually Exclusive and Exhaustive Events

FoundationHigherAQAEdexcel

Understand Mutually Exclusive and Exhaustive Events for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. Mutually exclusive events cannot happen together; exhaustive events cover all possible outcomes.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA and Edexcel specifications. Every worksheet comes with a full mark scheme.

Foundation

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Higher

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Topic overview

Two events are mutually exclusive if they cannot both happen at the same time. Rolling a 3 and rolling a 5 on a single dice are mutually exclusive, because one roll cannot be both.

For mutually exclusive events, \(P(A \text{ or } B) = P(A) + P(B)\). The probabilities are simply added, because there is no overlap between them.

A set of events is exhaustive if it covers every possible outcome, so one of them must happen. For a set of mutually exclusive and exhaustive events, the probabilities sum to exactly 1. Rolling 1, 2, 3, 4, 5 or 6 on a dice is both mutually exclusive and exhaustive.

Revision notes

Mutually exclusive events

Two events that cannot both happen at the same time.

Rolling a 3 and rolling a 5 on one dice are mutually exclusive. Rolling an even number and rolling a 4 are NOT, because rolling a 4 satisfies both.

Adding probabilities

For mutually exclusive events, \(P(A \text{ or } B) = P(A) + P(B)\).

The probabilities add because there is no overlap. If the events can both happen, simply adding would count the overlap twice and give too large an answer.

Exhaustive events

A set of events is exhaustive if it covers every possible outcome, so one must happen.

For a set that is both mutually exclusive and exhaustive, the probabilities sum to exactly 1. This provides a useful check on any probability calculation.

Key points

  • Mutually exclusive events cannot both happen.
  • For them, \(P(A \text{ or } B) = P(A) + P(B)\).
  • Probabilities add because there is no overlap.
  • Exhaustive events cover every outcome.
  • Mutually exclusive and exhaustive probabilities sum to 1.
  • Check whether events can overlap before adding.

Worked examples

Example 1

A dice is rolled. Work out the probability of getting a 2 or a 5. [2 marks]

Working

The events are mutually exclusive, so add: \(\frac{1}{6} + \frac{1}{6}\)identify that the probabilities can be added
\(= \frac{2}{6} = \frac{1}{3}\)work out and simplify

Example 2

Explain why rolling an even number and rolling a 4 are not mutually exclusive. [2 marks]

Working

Rolling a 4 is both an even number and a 4identify the overlap
so the two events can happen at the same time, which means they are not mutually exclusivestate the conclusion

Example 3

Three mutually exclusive and exhaustive events have probabilities 0.3, 0.45 and \(x\). Work out \(x\). [2 marks]

Working

The probabilities must sum to 1, so \(0.3 + 0.45 + x = 1\)use the exhaustive property
\(x = 1 - 0.75 = 0.25\)work out the missing probability

Common mistakes

  • Adding probabilities of events that overlap.

    Only mutually exclusive events can be added directly.

  • Confusing mutually exclusive with independent.

    Mutually exclusive means they cannot both happen; independent means one does not affect the other.

  • Saying exhaustive means the same as mutually exclusive.

    Exhaustive means all outcomes are covered; mutually exclusive means no overlap.

  • Not checking probabilities sum to 1.

    For a mutually exclusive exhaustive set they must.

Exam tips

  • Check whether events can overlap before adding.
  • Use the sum-to-1 property to find missing probabilities.
  • Keep mutually exclusive and independent distinct.
  • State which property you are using in your answer.

Key terms

Mutually exclusive
Events that cannot both happen.
Exhaustive
Covering every possible outcome.
Overlap
Outcomes belonging to both events.
Addition
Combining probabilities of mutually exclusive events.

Related topics

Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.