Addition and Multiplication of Probabilities (PS1 – 5.3) — Mathematics

Introduction

Combining probabilities is at the heart of almost every probability question on the 9709 exam. Once you can calculate individual probabilities (the prerequisite skill), the next step is deciding how to combine them when two or more events are involved. There are two fundamental rules: the addition rule (for "or" situations) and the multiplication rule (for "and" situations). Choosing the correct rule — and recognising the special conditions under which simplified versions apply — is what this topic is all about.

Core Concept

Events and Outcomes

Before combining probabilities, identify the events clearly. Two relationships between events matter most:

Mutually exclusive events: $A$ and $B$ cannot both occur at the same time. Knowing one has happened tells you the other has not.
Independent events: whether $A$ occurs has no effect on the probability that $B$ occurs.

These two properties are different — do not confuse them. Events can be neither, one, or both (the only case that is both is when one has probability 0).

The Addition Rule ("or")

Use addition when you want the probability that at least one of two events occurs (i.e. $A$ or $B$ or both).

General situation (not mutually exclusive):

P(A \text{ or } B) = P(A) + P(B) - P(\text{both } A \text{ and } B)

You subtract the overlap to avoid counting it twice.

Special case — mutually exclusive events ( $A$ and $B$ cannot both happen):

P(A \text{ or } B) = P(A) + P(B)

The overlap is zero, so nothing needs subtracting.

Exam note: The syllabus states that explicit use of the general formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ is not required. In 9709 questions you will either be told events are mutually exclusive (so use the simple version), or the problem structure will make the overlap obvious.

The Multiplication Rule ("and")

Use multiplication when you want the probability that both events occur (i.e. $A$ and $B$ ).

General situation (dependent events):

P(A \text{ and } B) = P(A) \times P(B \mid A)

where $P(B \mid A)$ is the conditional probability of $B$ given that $A$ has already occurred.

Special case — independent events (one event does not affect the other):

P(A \text{ and } B) = P(A) \times P(B)

Because independence means $P(B \mid A) = P(B)$ .

Choosing the Right Rule — A Quick Guide

Keyword / situation	Rule to use
"or", "at least one of", "either"	Addition
Mutually exclusive events	$P(A) + P(B)$
"and", "both", sequence of events	Multiplication
Independent events	$P(A) \times P(B)$
Dependent events (e.g. without replacement)	$P(A) \times P(B \mid A)$

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