CS205: Building with Artificial Intelligence

Events A and B are defined to be statistically independent if the probability of the intersection of A and B is equal to the product of the probabilities of A and B:

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

If P(B) is not zero, then this is equivalent to the statement that

$P(A\mid B)=P(A)$.

Similarly, if P(A) is not zero, then

$P(B\mid A)=P(B)$

is also equivalent. Although the derived forms may seem more intuitive, they are not the preferred definition as the conditional probabilities may be undefined, and the preferred definition is symmetrical in A and B. Independence does not refer to a disjoint event.

It should also be noted that given the independent event pair [A B] and an event C, the pair is defined to be conditionally independent if the product holds true:

$P(AB\mid C)=P(A\mid C)P(B\mid C)$

This theorem could be useful in applications where multiple independent events are being observed.

Independent events vs. mutually exclusive events

The concepts of mutually independent events and mutually exclusive events are separate and distinct. The following table contrasts results for the two cases (provided that the probability of the conditioning event is not zero).