Basic Notions of Probability Theory - Probabilistic Composition of Preferences, Theory and Applications - page 119

Intuitively, independence means that the occurrence of B does not change the
probability of A occurring or not. An equivalent de nition of independence of

events, simpler but less intuitive is:

A and B are independent $ PA ð \ B Þ ¼ PA ðÞ PB ðÞ:

nition has the advantage of being applicable also to pairs of events of null
probability. By this extension, events of null probability are independent of any other.
It is useful to consider the question about the independence of the members of a
partition. They are not independent of each other (except those of null probability,
which are unimportant for the applications of the Total Probability Theorem). In
fact two disjoint events of nonnull probability are never independent of each other,
as their conditional probabilities are null.
Another basic theorem, the Bayes Theorem, is used to discover the probability of
each element of a partition conditionally on an event A that may be found to have
occurred when is known the probability of A conditional on the events of the
partition.

This de

Bayes Theorem

pB ð j A Þ ¼ pA ð j B Þ pB ðÞ= pA ðÞ;

for any pair of events A and B.
In this context, p(B|A) is called the posterior probability of B and p(B) is called
its prior probability.
This result has important practical consequences for statistical inference. One of
the most important is to call attention to the importance of correctly evaluating prior
probabilities.
Consider, for instance, the case of an event B of low prior probability such as an
individual carrying the virus of a rare disease. That means, B is the set of members
of a population S with the rare disease. Suppose a very accurate blood test is
designed to detect the presence of such virus. Let A be the event that an individual
in the population is pointed by this blood test as infected by the virus, that means, A
is the set of the elements of S for which an application of the blood test gives a
positive result.
To make things more concrete, let us suppose p(A|B) = 0.99, p(A|B c ) = 0.001
and p(B) = 0.0001. In that case, this prior being correct, you should not be very
worried if the test points you as having the disease. Even though the probability of
the test offering a wrong result is of only

pB C ¼

pA c j B C ð B Þ pB ðÞþ pA j 0 : 01 0 : 0001 þ 0 : 001 0 : 9999

¼ 0 : 0010009

and the probability of it presenting the positive result that you received is of only

pB C ¼ B C

pA ðÞ ¼ pA ð j B Þ pB ðÞþ pA j 0 : 99 0 : 0001 þ 0 : 001 0 : 9999

¼ 0 : 0010989 :