Mutually Exclusive Events: KS3 Maths

What does mutually exclusive mean?

Two events are mutually exclusive (also called disjoint) if they share no outcomes. If one event happens, the other cannot happen in the same trial.

Examples of mutually exclusive events:

• Rolling an even number and rolling an odd number on one die — a number cannot be both.
• Drawing a red card and drawing a club from one card — a red club does not exist.
• A spinner landing on blue and landing on yellow in a single spin.

Examples that are NOT mutually exclusive:

• Drawing a king and drawing a heart — the king of hearts satisfies both.
• Rolling a number greater than 4 and rolling an even number — 6 satisfies both.

When two events CAN happen at the same time, they are called compatible (not mutually exclusive).

How do you recognise mutually exclusive events from a list of outcomes?

Write out the outcomes for each event and look for any shared outcomes. If the two lists have no outcome in common, the events are mutually exclusive.

Example: A fair die is rolled once.

• Event A: rolling a prime number → outcomes {2, 3, 5}.
• Event B: rolling a multiple of 4 → outcomes {4}.

The two sets share no outcome, so A and B are mutually exclusive.

Counter-example:

• Event C: rolling an even number → {2, 4, 6}.
• Event D: rolling a prime number → {2, 3, 5}.

Both sets contain 2, so C and D are not mutually exclusive.

What is the addition rule for mutually exclusive events?

When A and B are mutually exclusive:

P(A or B) = P(A) + P(B)

You simply add the two probabilities. This works because the events share no outcomes — there is no risk of counting anything twice.

Worked example: A fair die is rolled once. Find the probability of scoring a 2 or a 5.

1. P(2) = 1/6, P(5) = 1/6.
2. Rolling a 2 and rolling a 5 are mutually exclusive (only one number per roll).
3. P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3.

Warning: You can ONLY use P(A or B) = P(A) + P(B) when the events are mutually exclusive. If they can overlap, a different formula is required (taught at GCSE).

What are exhaustive events?

A set of events is exhaustive if at least one of them must always happen — together they cover every possible outcome.

For a fair die, the set {1, 2, 3, 4, 5, 6} is exhaustive because every roll must produce one of those values.

Mutually exclusive AND exhaustive events are especially useful: they split all outcomes into separate groups that together account for everything, and their probabilities always sum to exactly 1.

Example: When flipping a coin, {Heads, Tails} is mutually exclusive and exhaustive. P(H) + P(T) = 0.5 + 0.5 = 1. ✓

How does the complement rule connect to mutually exclusive events?

An event A and its complement A′ ("not A") are always both mutually exclusive AND exhaustive. They cannot both happen, and together they cover everything.

This gives the complement rule:

P(A′) = 1 − P(A)

Example: The probability that a bus arrives on time is 0.72. What is the probability it does NOT arrive on time?

P(not on time) = 1 − 0.72 = 0.28.

The complement rule is really the addition rule applied to the simplest mutually exclusive exhaustive pair: event and its opposite.

How do you use these rules in probability problems?

Worked example: A bag contains coloured tiles. The probability of drawing red is 0.35, the probability of drawing blue is 0.25. No tile is both colours. Find the probability of drawing red or blue, and the probability of drawing neither.

1. Red and blue are mutually exclusive (a tile cannot be two colours at once).
2. P(red or blue) = 0.35 + 0.25 = 0.60.
3. P(neither red nor blue) = 1 − 0.60 = 0.40.

Frequently asked questions

Can more than two events be mutually exclusive?

Yes. A set of events is mutually exclusive if no two of them can occur in the same trial. For a single die roll, {1, 2, 3, 4, 5, 6} are all mutually exclusive with each other. You can add as many probabilities as you like: P(1 or 2 or 3) = 1/6 + 1/6 + 1/6 = 1/2.

Is every pair of events either mutually exclusive or exhaustive?

No — these are separate properties. Two events can be neither. For example, rolling an even number and rolling a prime on a die: they are not mutually exclusive (2 is both), and not exhaustive (1 is neither). Always check each property independently.

Does P(A or B) = P(A) + P(B) always work?

Only when A and B are mutually exclusive. If they share outcomes, adding would count the shared outcomes twice. At GCSE the full formula is P(A or B) = P(A) + P(B) − P(A and B), which adjusts for any overlap. At KS3, the simpler addition rule is used exclusively for mutually exclusive events.

How do I know if a question is testing mutually exclusive events?

Look for phrasing such as "the probability of A or B" where the context makes clear only one can happen. Keywords include "or", "either … or", and statements like "these events cannot both occur". If the problem gives a list of probabilities that sum to less than 1, it may also be asking you to find the probability of the remaining outcome.

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Professor Pi can walk you through probability problems one step at a time — visit aitutors.me.