What Does It Mean for Two Events to Be Mutually Exclusive?

Two events are mutually exclusive if they cannot occur in the same trial. Therefore, their intersection is empty and $P(A \cap B)=0$.

For example, suppose a fair six-sided die is rolled once. Let $A$ be the event “an even number is rolled” and $B$ be the event “a 5 is rolled.” Since no outcome is both even and 5, $A$ and $B$ are mutually exclusive.

On a Venn diagram, the circles representing $A$ and $B$ have no overlapping region. In set notation, $A \cap B=\varnothing$. This visual interpretation matches the probability condition above.

In IB Mathematics: Analysis and Approaches SL 4.6, the combined-events rule is

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

The subtraction term prevents outcomes in the intersection from being counted twice. For mutually exclusive events, the intersection is empty, so there is no double-counting to correct and the rule simplifies to

$$P(A \cup B)=P(A)+P(B).$$

Using the die example,

$$P(A \cup B)=\frac{3}{6}+\frac{1}{6}=\frac{2}{3}.$$

Always check the actual outcome sets before using this simplified addition rule in a probability calculation.

The events do not need to include every possible outcome. Mutually exclusive means only that they do not overlap. If mutually exclusive events together include every possible outcome, they are also exhaustive.

A common misconception is that mutually exclusive events are independent. They are generally not: if $A$ occurs, a mutually exclusive event $B$ becomes impossible. They can be both mutually exclusive and independent only if at least one event has probability zero.

Exam technique: This SL 4.6 concept can appear on AA Paper 1 or Paper 2. To justify that events are mutually exclusive, explicitly state that $A \cap B=\varnothing$ or $P(A \cap B)=0$; do not merely say that the events are “different.”