GCSE Mathematics 05 — Probability

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Topics include The Probability Scale and Basic Definitions, Theoretical Probability, Experimental Probability & Relative Frequency, Mutually Exclusive Events (The OR Rule), Independent Events (The AND Rule), Combined Events & Inclusion-Exclusion, Sample Spaces & Systematic Listing, and Set Notation & Venn Diagrams.

Mathematics EN

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Mutually Exclusive Events (The OR Rule)

Handling events that cannot happen simultaneously using the Addition Rule.

Key points

Mutually Exclusive: Events that cannot happen at the same time (Intersection is zero).
Addition Rule: $P(A \text{ or } B) = P(A) + P(B)$
Mutually exclusive ⇒ $P(A\cap B)=0$ so $P(A\text{ and }B)=0$ . The multiplication rule $P(A\cap B)=P(A)P(B)$ is for independent events (or when independence is stated/assumed).
Complement Rule: $P(\text{Not } A) = 1 - P(A)$
For exhaustive events (covering all possibilities), the sum of probabilities is 1.

Worked example

Question

A bag contains red, blue, and green balls.

P(\text{Red}) = 0.3

and

P(\text{Blue}) = 0.45

. Calculate

P(\text{Green})

Solution

1. Identify that picking a specific colour is mutually exclusive.
2. Sum the known probabilities:

0.3 + 0.45 = 0.75

3. Subtract from 1 to find the remaining probability:

P(\text{Green}) = 1 - 0.75 = 0.25

Common pitfalls

Adding probabilities for non-mutually exclusive events (e.g., P(King) + P(Red) in a deck of cards) without subtracting the overlap.
Confusing 'Mutually Exclusive' with 'Independent'.
Arithmetic errors when subtracting from 1.
Using $P(A)P(B)$ for mutually exclusive events. Unless one probability is 0, this gives a non‑zero value even though $P(A\text{ and }B)=0$ .

Prerequisites

Basic addition/subtraction of decimals
Understanding sets

Further resources

Additional study resource
Supplementary resource.

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