Independence and mutual exclusivity describe different relationships between events. Mutual exclusivity is about whether two events can happen together, and independence is about whether learning that one event occurred changes the probability of the other. For events with positive probability, mutually exclusive events cannot be independent.

Mutual exclusivity

Events \(A\) and \(B\) are mutually exclusive when \(P(A\cap B)=0\). They have no overlap, and if one occurs, the other did not. This condition simplifies the addition rule: \(P(A\cup B) = P(A)+P(B)\) when the events are mutually exclusive.

Independence

Events \(A\) and \(B\) are independent when \(P(A\mid B)=P(A)\). Knowing that \(B\) occurred gives no new information about the probability of \(A\), and an equivalent condition is \(P(A\cap B) = P(A)P(B)\). Independence is therefore a numerical relationship among probabilities.

Watch the overlap change

The events below have fixed probabilities \(P(A)=0.5\) and \(P(B)=0.4\).

The rectangle is the sample space; the two regions are the events, and the dark region is their overlap. Independence requires the overlap to equal P(A) times P(B), which is 0.20 here, marked on the slider's scale by the readout. Slide fully left to make the events mutually exclusive and watch the conditional probability P(A given B) fall to zero, as far from P(A) = 0.5 as it can be. Slide to 0.20 and the conditional probability matches P(A) exactly: knowing B tells you nothing about A. Independence is that single point on the slider; exclusivity is the far end of it.

Independence requires \(P(A\cap B) = (0.5)(0.4) = 0.20\), and at overlap 0.20, \(P(A\mid B)=P(A)=0.5\). Move the overlap to zero, and the events are now mutually exclusive. Then \(P(A\mid B)=0\), which is very different from \(P(A)=0.5\).

Learning that \(B\) happened tells us with certainty that \(A\) did not. That is dependence, not independence.

An overlapping pair can be independent

Draw one card from a standard deck, let \(A\) be the event that the card is a heart, and let \(B\) be the event that the card is a king. The events overlap because the king of hearts belongs to both. But \(P(A\cap B) = \tfrac1{52}\), and \(P(A)P(B) = \tfrac{13}{52}\cdot\tfrac4{52} = \tfrac1{52}\), so the events are independent.

Overlap and independence are compatible. No overlap and independence are not compatible when both events have positive probability.

Two-way tables

In a two-way table, independence can be checked by comparing a conditional proportion with the corresponding marginal proportion. For example, compare \(P(A\mid B)\) with \(P(A)\), or use the multiplication condition \(P(A\cap B)=P(A)P(B)\).

The wording of the categories does not determine independence. The probabilities do.