Introduction to Probability and Statistics for Engineers and Scientists

3.7Bayes’ Formula 75

=

(.6)(.9)

(.9)(.6)+(.2)(.4)

=

54

62

=.871

which is slightly less than in the previous case (why?). ■

Equation 3.7.1 may be generalized in the following manner. Suppose thatF 1 ,F 2 ,...,Fn
are mutually exclusive events such that

⋃n

i= 1

Fi=S

In other words, exactly one of the eventsF 1 ,F 2 ,...,Fnmust occur. By writing

E=

⋃n

i= 1

EFi

and using the fact that the eventsEFi,i=1,...,nare mutually exclusive, we obtain that

P(E)=

∑n

i= 1

P(EFi)

=

∑n

i= 1

P(E|Fi)P(Fi) (3.7.2)

Thus, Equation 3.7.2 shows how, for given eventsF 1 ,F 2 ,...,Fnof which one and only
one must occur, we can computeP(E) by first “conditioning” on which one of theFi
occurs. That is, it states thatP(E) is equal to a weighted average ofP(E|Fi), each term
being weighted by the probability of the event on which it is conditioned.
Suppose now thatEhas occurred and we are interested in determining which one of
Fjalso occurred. By Equation 3.7.2, we have that

P(Fj|E)=

P(EFj)

P(E)

=

P(E|Fj)P(Fj)

∑n

i= 1

P(E|Fi)P(Fi)

(3.7.3)

Equation 3.7.3 is known asBayes’ formula, after the English philosopher Thomas Bayes. If
we think of the eventsFjas being possible “hypotheses” about some subject matter, then