Correlation between events

Correlation measures how much the occurrence of one event changes the probability of another, comparing the probability of the intersection with the one we would have if the two events were independent.

Definition — Correlation

The correlation between two events $A$ and $B$ is the quantity: $\operatorname{corr}(A,B) = \frac{P(A \cap B)}{P(A) \cdot P(B)} - 1.$

• If $\operatorname{corr}(A,B) = 0$: the events are independent (a definition equivalent to $P(A \cap B) = P(A)P(B)$).
• If $\operatorname{corr}(A,B) > 0$: positive correlation. The occurrence of $B$ favours $A$, that is, $P(A \mid B) > P(A)$.
• If $\operatorname{corr}(A,B) < 0$: negative correlation. The occurrence of $B$ works against $A$, that is, $P(A \mid B) < P(A)$.

Example — Independent vs correlated events

I roll two fair dice. Event $A$: “the first die shows 6”. Event $B$: “the second die shows 6”. $P(A) = P(B) = 1/6$, $P(A \cap B) = 1/36$. The product $P(A) \cdot P(B) = 1/36$ coincides with $P(A \cap B)$: independent. ✓

Event $C$: “the sum of the two dice is $12$”. $P(C) = 1/36$ (only the outcome $(6,6)$). $P(C \mid A)$: knowing that the first is $6$, the second must be $6$: $P(C \mid A) = 1/6$. Since $1/6 > 1/36$, the two events $A$ and $C$ are positively correlated.

Links

Topics: Probability
Concepts: Correlation · Independent events · Conditional probability
Skills: Calculating probability