P
A
c
P
B

A
c
and shows that, for any event
A
with 0
P
A
1, we can write
P
B
as a weighted average of the two conditional probabilities
P
B

A
and
P
B

A
c
, with respective weights
P
A
and
P
A
c
1
−
P
A
.
63
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∙
Except by fluke, it is not true that
P
B

A
P
B

A
c
1. [By
contrast, it is true that
P
B

A
P
B
c

A
1 always.]
∙
To get Bayes’ rule, we use the previous expression for
P
B
:
P
A

B
P
B

A
P
A
P
B

A
P
A
P
B

A
c
P
A
c
∙
In other words, we can obtain
P
A

B
by knowing
P
B

A
,
P
B

A
c
,
and
P
A
.
64
EXAMPLE
: Deciding whether to locate a business at a particular site.
Want to know the chances that the business will be profitable. Can
conduct a study that provides input.
A
site is profitable
B
study says site is profitable
65
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∙
Suppose 32% of all similar sites are profitable, so
P
A
.32. Based
on past experience where studies were conducted, also know
P
B

A
.65 and
P
B

A
c
.29. Using Bayes’ rule,
P
A

B
.65
.32
.65
.32
.29
.68
≈
.513
∙
In other words, if the study says the site will be profitable, the site
actually will be profitable with probability .513. This is notably higher
than the unconditional probability – that is, the probability without
knowing the outcome of the study – which is .32.
∙
What is
P
A

B
c
?
66