Exclusive and Independent Events (9709 PS1 – 5.3)

Core Concept

Mutually Exclusive Events

Two events $A$ and $B$ are mutually exclusive (also called exclusive) if they cannot both occur at the same time. In set notation, their intersection is empty:

On a Venn diagram the two circles do not overlap. Because they share no outcomes, the addition rule simplifies to:

Key idea: If one event happens, the other is impossible.

Independent Events

Two events $A$ and $B$ are independent if the occurrence of one gives no information about whether the other occurs — knowing $A$ happened does not change the probability of $B$, and vice versa.

The formal test for independence is:

If this equation holds, $A$ and $B$ are independent; if it fails, they are dependent.

This links directly to conditional probability (a prerequisite idea): $A$ and $B$ are independent if and only if $P(A \mid B) = P(A)$ and $P(B \mid A) = P(B)$, but the 9709 syllabus tests independence by comparing $P(A \cap B)$ with $P(A) \times P(B)$ — that is the required method.

The Crucial Distinction

Two non-trivial events cannot be both mutually exclusive AND independent. If $P(A \cap B) = 0$ but $P(A) > 0$ and $P(B) > 0$, then $P(A) \times P(B) > 0 \neq 0$, so independence fails.