Mutually Exclusive Events –

Mutually exclusive events are a core probability concept tested on both Foundation and Higher GCSE Maths papers across AQA, Edexcel and OCR. Understanding when events cannot happen at the same time allows you to add probabilities — a rule that underpins many exam questions. You also need to know the complement rule and what exhaustive events are. This guide covers all the essentials with clear worked examples and practice questions. For an overview of every topic, see our complete GCSE Maths topics list.

What Are Mutually Exclusive Events?

Two events are mutually exclusive if they cannot happen at the same time. For example, when you flip a coin, it lands on either heads or tails — not both. Rolling a 3 and rolling a 5 on a single dice are mutually exclusive.

Key Formulas

The addition rule only works when A and B cannot happen together. If events can overlap, this formula does not apply.

Exhaustive Events

Events are exhaustive if they cover all possible outcomes. For exhaustive events, the sum of all probabilities equals 1. For example, the outcomes 1, 2, 3, 4, 5 and 6 on a dice are exhaustive — they cover every possibility.

Combining Both Concepts

If events are both mutually exclusive and exhaustive, their probabilities add up to exactly 1. This is why P(not A) = 1 − P(A) works — event A and "not A" are mutually exclusive and exhaustive.

Step-by-Step Method

1. Identify whether the events can happen at the same time.
2. If they cannot (mutually exclusive), add the probabilities: P(A or B) = P(A) + P(B).
3. If you need the probability of something not happening, use P(not A) = 1 − P(A).
4. If given several probabilities that should cover all outcomes, check they sum to 1.

Worked Example 1 — Foundation Level

Question: A bag contains 10 counters: 3 red, 4 blue and 3 green. A counter is picked at random. Find the probability of picking a red or green counter.

Working:

Red and green are mutually exclusive — a counter cannot be both red and green.

P(red) = 3/10. P(green) = 3/10.

P(red or green) = 3/10 + 3/10 = 6/10 = 3/5.

Answer: The probability of picking a red or green counter is 3/5.

Worked Example 2 — Higher Level

Question: The probability that a student walks to school is 0.35, takes the bus is 0.25 and cycles is 0.15. These are the only three ways students travel. (a) Find P(walks or cycles). (b) Find P(goes by car).

Working:

(a) Walking and cycling are mutually exclusive. P(walks or cycles) = 0.35 + 0.15 = 0.50.

(b) The three given modes are not exhaustive — some students go by car. Total of given = 0.35 + 0.25 + 0.15 = 0.75. P(car) = 1 − 0.75 = 0.25.

Answer: (a) 0.50. (b) 0.25.

Worked Example 3 — Exam Style

Question: A spinner has sections labelled A, B, C and D. The table shows some probabilities.

(a) Find x. (b) Find P(A or D). (c) Find P(not B).

Working:

(a) All outcomes are exhaustive, so probabilities sum to 1: 0.1 + 0.35 + x + 0.2 = 1. 0.65 + x = 1. x = 0.35.

(b) A and D are mutually exclusive. P(A or D) = 0.1 + 0.2 = 0.3.

(c) P(not B) = 1 − 0.35 = 0.65.

Answer: (a) x = 0.35. (b) P(A or D) = 0.3. (c) P(not B) = 0.65.

Common Mistakes

• Adding probabilities for non-mutually-exclusive events. If events can happen at the same time (e.g. picking a red card and picking a king), you cannot simply add — you would double-count the overlap.
• Probabilities not summing to 1. If a question gives all possible outcomes, check that your probabilities add to 1. If they do not, you have made an error.
• Confusing "or" with "and." "Or" means either one (use addition for mutually exclusive). "And" means both together (use multiplication for independent — see the next topic).

Exam Tips

• "Or" in probability usually means add (for mutually exclusive events).
• If you are given all probabilities except one, subtract the total of the known probabilities from 1.
• State clearly whether events are mutually exclusive to justify using the addition rule.
• At Higher level, non-mutually-exclusive events need the formula P(A or B) = P(A) + P(B) − P(A and B) — but this is covered in Venn diagrams.
• For combined events, see independent and dependent events. For key formulas, visit our GCSE Maths formulas page.

Practice Questions

Q1 (Foundation): A dice is rolled. Find the probability of getting a 2 or a 5.

Q2 (Foundation): The probability of picking a red sweet is 0.4. What is the probability of not picking a red sweet?

Q3 (Higher): Events X, Y and Z are mutually exclusive and exhaustive. P(X) = 0.3 and P(Y) = 0.45. Find P(Z) and P(X or Z).

Practise mutually exclusive events and probability rules free on GCSEMathsAI.

Related Topics

• Independent and Dependent Events
• Probability Scale and Basic Probability
• Venn Diagrams

Summary

• Mutually exclusive events cannot happen at the same time.
• For mutually exclusive events, P(A or B) = P(A) + P(B).
• Exhaustive events cover all possible outcomes and their probabilities sum to 1.
• The complement rule P(not A) = 1 − P(A) applies because A and "not A" are mutually exclusive and exhaustive.
• Do not add probabilities if events can overlap — this only works for mutually exclusive events.