Assume now that someone develops a diagnostic test for lurgi, but it is not
completely accurate. Let B denote the event that the test shows the disease is
present. Assume that if the person has the disease that the evidence clearly
shows that the probability of the test indicating the presence of the disease is
0.90. Assume that if the person actually does not have the disease but the test
indicates that the disease is present is 0.15. Using the standard notation these
probabilities can be written as follows:
P(B|A1) = 0.90
P(B|A2) = 0.15
The point to Bayes Theorem is that it enables us to upgrade the relevant
probabilities. To illustrate, assume that a person X is selected at random and
the diagnostic test is performed. The test indicates that the disease is present.
What is the probability that the person actually has the disease? In symbolic
form, we want to know P(A1|B). This is called a posterior or revised
probability, because it is revised following the discovery of additional
information since the original prior probability was determined. In this case
where there are only two possible events, having the disease (A1) or not having
the disease (A2), the probability is given by the formula:
P(A1|B) =
)
A2
|
P(B
x
)
P(A2
)
A
|
P(B
x
)
P(A
)
A
|
P(B
x
)
P(A
1
1
1
1
In this illustration there were only two events (having the disease (A1) or not
having the disease (A2)) Obviously, it is possible that there can be more than
two possible events. In this case the denominator needs to be adjusted. The
formula now becomes:
P(A1|B) =
)
A
|
P(B
x
)
P(A
)
A
|
P(B
x
)
P(A
)
A1
|
P(B
x
)
P(A1
)
A1
|
P(B
x
)
P(A
n
n
2
2
1
Calculations for the answer to the question are set out in the following table:
Events
Prior
Probability
Conditional
Probability
Joint
Probability
Posterior
Probability
A1
P(A1)
P(B|A1)
P (A1 and B)
P(A1|B)
Disease
0.05
0.90
0.0450
0.0450/.1875 = 0.24
No Disease
0.95
0.15
0.3425
0.1425/.1875 = 0.76
P(B) = 0.1875
Total 1.00
Figure 9.6 Calculation of Probabilities
The final column “Posterior Probability” gives the answer. The probability that
X has the disease is 0.24 or 24%, while the probability that X does not have
the disease is 0.76 or 76%
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