Independence versus mutually exclusive

◈ 5 cards

Three equivalent tests for independence, run on every pair; and why two disjoint events with positive probability must be dependent.

What independence means

Two events are independent when knowing one occurred tells you nothing about the other: $P (A ∣ B) = P (A)$ . Substitute into the multiplication rule and two more forms fall out. Any one of the three is a test; all three agree:

$P (A ∣ B) = P (A)$
$P (B ∣ A) = P (B)$
$P (A \cap B) = P (A) P (B)$

The third is the one to run when the paper gives probabilities: multiply the two marginals and compare with the joint. Assume dependence until a test shows otherwise — independence is a property to be checked, not a default.

Worked example — are late and error independent?

From Maple Ledger's table, $P (late) P (error) = 0.24 \times 0.16 = 0.0384$ , but $P (late \cap error) = 0.06$ . Not equal, so dependent. The first test says the same: $P (error ∣ late) = 0.25 \neq = P (error) = 0.16$ — a late invoice is more likely to carry an error than an invoice in general. Any one test settles it; run whichever the given numbers make easiest.

A pair that is independent

Georgian Bay Mutual finds that 40 % of its auto claims involve a rental car ( $A$ ) and 50 % are filed online ( $B$ ), with 20 % both. $P (A) P (B) = 0.4 \times 0.5 = 0.20 = P (A \cap B)$ — independent. For independent events the multiplication rule simplifies to $P (A \cap B) = P (A) P (B)$ , and the addition rule then gives $P (A \cup B) = 0.4 + 0.5 - 0.2 = 0.7$ .

Mutually exclusive is not independent

The two phrases sound alike and are opposites in the case that matters. Take a claim that is a total loss ( $T$ , probability 0.30) or a partial loss ( $R$ , 0.25), never both. Run the product test: $P (T) P (R) = 0.075$ , but $P (T \cap R) = 0$ . Unequal — dependent. In words: if you learn the claim was a total loss, you know for certain it was not a partial one. Knowing $T$ changed $P (R)$ from 0.25 to 0. That is dependence in its strongest form.

The only way disjoint events pass the product test is if one of them has probability zero. So: two mutually exclusive events with positive probabilities are always dependent. The paper asks this, in some disguise, every year.

Keeping the two apart

	Mutually exclusive	Independent
Definition	cannot both occur	one tells nothing about the other
Test	$P (A \cap B) = 0$	$P (A \cap B) = P (A) P (B)$
Effect on rules	addition rule loses its last term	multiplication rule loses its condition

One is about the overlap being empty; the other is about the overlap being exactly what the margins predict.