Mutually exclusive events and independence

Two different ideas that sound similar

• Mutually exclusive and independent are both named in the explanatory notes, they both concern a pair of events, and they mean completely different things.
• Confusing them is the most common conceptual error at this level.

	Mutually exclusive	Independent
Meaning	They cannot both happen	One happening does not change the chance of the other
Test	$P(A \text{ and } B) = 0$	$P(A \text{ and } B) = P(A) \times P(B)$
Venn diagram	Circles do not overlap	Circles do overlap (in general)
Addition rule becomes	$P(A \text{ or } B) = P(A) + P(B)$	No simplification
Example	A single die shows a 3 / shows a 5	First die shows 3 / second die shows 5

Mutually exclusive events

• Events are mutually exclusive if they cannot occur together:

$$P(A \text{ and } B) = 0$$

• Because the overlap is zero, the addition rule simplifies:

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

• Examples: a person is aged 15–19 or aged 20–24; a single card is a heart or a spade; a customer pays by cash or by card (if only one method is allowed).
• Exhaustive is a different word again: a set of events is exhaustive if between them they cover the whole sample space. A set that is both mutually exclusive and exhaustive is a partition, and its probabilities sum to 1.

Independent events

• Events are independent if knowing one occurred does not change the probability of the other:

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

• The multiplication form is the practical test. To show two events are independent, compute $P(A) \times P(B)$ and compare it with $P(A \text{ and } B)$. If they are equal, the events are independent; if not, they are not.
• Examples: two tosses of a coin; the weather in Invercargill today and the result of a rugby match in Hamilton; drawing a card with replacement.

Why mutually exclusive events are not independent

• This catches almost everyone, so reason it through once and remember the argument.
• Suppose $A$ and $B$ are mutually exclusive with $P(A) > 0$ and $P(B) > 0$.
• If you know $B$ has happened, then $A$ cannot have happened, so $P(A \mid B) = 0$.
• But $P(A) > 0$. So $P(A \mid B) \neq P(A)$, and the events are dependent.

Mutually exclusive events are about as dependent as two events can be — one occurring tells you with certainty that the other did not.

Checking independence from data

• Given a two-way table or a set of probabilities, the procedure is fixed:
  • Compute $P(A)$, $P(B)$ and $P(A \text{ and } B)$.
  • Compare $P(A) \times P(B)$ with $P(A \text{ and } B)$.
  • Equal → independent. Not equal → not independent, and the direction tells you more:
    • $P(A \text{ and } B) > P(A)P(B)$ → the events are positively associated (each makes the other more likely)
    • $P(A \text{ and } B) < P(A)P(B)$ → negatively associated.
• Always state the conclusion in context, and where the numbers come from data, note that small departures may reflect sampling variability rather than genuine dependence.

Independence in a real context is an assumption, not a fact

• Questions frequently ask whether an independence assumption is reasonable. Look for a shared cause:
  • Two people in the same household catching an illness — not independent.
  • Two flights on the same day in the same weather — not independent.
  • Two components from the same production batch failing — not independent.
• The test to apply: is there anything that would affect both at once? If yes, they are dependent.