Conditional probability

Probability of an event occurring, given that another event has already occurred
Thus the equations can be combined to find a new representation of the :

The Borel–Kolmogorov paradox demonstrates this with a geometrical argument.

Suppose that somebody secretly rolls two fair six-sided dice, and we wish to compute the probability that the face-up value of the first one is 2, given the information that their sum is no greater than 5.

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).

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).

These fallacies should not be confused with Robert K. Shope's 1978 , which deals with counterfactual examples that beg the question.