Probability theory provides a robust and well-understood platform to handle uncertainty. In addition to "prior" probability, it is also useful to master conditional probability to sharpen our ability to reason about uncertain events. Can you explain how conditional probability works and how to analyze the likelihood of events with some apparent dependence on one another?
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:
.
If P(B) is not zero, then this is equivalent to the statement that
.
Similarly, if P(A) is not zero, then
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:
This theorem could be useful in applications where multiple independent events are being observed.
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).
| If statistically independent | If mutually exclusive | |
|---|---|---|
| P ( A ∣ B ) = |
P ( A ) |
0 |
| P ( B ∣ A ) = |
P ( B ) |
0 |
| P ( A ∩ B ) = |
P ( A ) P ( B ) |
0 |
In fact, mutually exclusive events cannot be statistically independent (unless both of them are impossible), since knowing that one occurs gives information about the other (in particular, that the latter will certainly not occur).