Equation (3.1) may be generalized in the following manner: Suppose that
are mutually exclusive events such that
In other words, exactly one of the events
must occur. By
and using the fact that the events
Thus Equation (3.3) shows how, for given events
one and only one must occur, we can compute P(E) by first conditioning on
which one of the Fi occurs. That is, Equation (3.3) states that P(E) is
equal to a weighted average of P(E|Fi), each term being weighted by the
probability of the event on which it is conditioned.
Suppose now that E has occurred and we are interested in determining which
one of the Fj also occurred. By Equation (3.3), we have the following
Equation (3.4) is known as Bayes' formula, after the English philosopher
Thomas Bayes. If we think of the events Fj as being possible "hypotheses"
about some subject matter, then Bayes' formula may be interpreted as showing
us how opinions about these hypotheses held before the experiment
[that is, the P(Fj)] should be modified by the evidence of the experiment.