Mutually exclusive events and adding probabilities

8 minAS Level5.3.25.3.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can find $P(A \text{ or } B)$ , recognise when two events are mutually exclusive, and explain why the overlap must be subtracted when they are not.

What "or" means in probability

In a probability question, "A or B" means A happens, or B happens, or both happen. It is the event that at least one of them occurs.

That "or both" is easy to miss, and it is the whole difficulty of this topic. If A and B can never happen together, the arithmetic is easy: just add.

If they can happen together, adding double-counts the shared outcomes, and we must correct for it.

Definition

Mutually exclusive events

Two events are mutually exclusive if they cannot both happen in the same trial. (A trial is one go of the experiment, such as one roll of a die.)

Formally, $A$ and $B$ are mutually exclusive when $P(A \cap B) = 0$ , where $A \cap B$ means "A and B both happen".

For mutually exclusive events, $P(A \cup B) = P(A) + P(B)$ , where $A \cup B$ means "A or B".

Why the overlap must be subtracted

Think of the sample space as a tray of outcomes and each event as a region drawn round its outcomes.

Adding $P(A)$ and $P(B)$ measures the two regions one after the other. Any outcome sitting in both regions gets measured twice, so the total comes out too large by exactly the probability of the overlap.

Subtracting $P(A \cap B)$ once repairs that:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

The syllabus does not require you to use this formula — you may always list the outcomes in $A \cup B$ instead — but it is much faster once you trust it.

Worked example: Example 1: mutually exclusive, so just add

A fair six-sided die is rolled once. Event $A$ is "the score is 2" and event $B$ is "the score is 5". Find $P(A \text{ or } B)$ .

Have a go first?

Worked example: Example 2: not exclusive, so subtract the overlap

A fair six-sided die is rolled once. Event $A$ is "the score is even", so $A = \{2, 4, 6\}$ , and event $B$ is "the score is greater than 4", so $B = \{5, 6\}$ . Find $P(A \text{ or } B)$ .

Have a go first?

Check 1 — addition in action

CalculateQuestion 1 · Apply

Answer the 3 questions above to continue.