Mutually exclusive and independent events

Two words that are easy to confuse, but which describe entirely different situations, are “mutually exclusive” and “independent”.

Definition — Mutually exclusive and independent events

• Two events $A$ and $B$ are said to be mutually exclusive (or incompatible) if they cannot occur at the same time, that is, $P(A \cap B) = 0$.
• Two events $A$ and $B$ are said to be independent if the occurrence of one does not affect the probability of the other, that is, $P(A \cap B) = P(A) \cdot P(B).$

Careful

Do not confuse “mutually exclusive” with “independent”! They are opposite concepts: if $A$ and $B$ are mutually exclusive (and both have non-zero probability), then $P(A \cap B) = 0 \neq P(A) \cdot P(B)$, so they are not independent — indeed, knowing that $A$ has occurred rules out $B$.

Example — Checking for independence

Let $P(A) = 0,2$, $P(B) = 0,2$, $P(A \cap B) = 0,1$. Are they independent?

We compute $P(A) \cdot P(B) = 0,2 \cdot 0,2 = 0,04$, whereas $P(A \cap B) = 0,1$. Since $0,1 \neq 0,04$, the events are not independent.

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Topics: Probability
Concepts: Mutually exclusive events · Independent events · Probability
Skills: Calculating probability