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Assessing the value of new information 231
this indication using the probability tree in Figure 8.10(a). The prior
probability of the virus being present and the test correctly indicating
this is 0.3 × 0.9, which is 0.27. Similarly, the probability of the virus being
absent and the test incorrectly indicating its presence is 0.7 × 0.2, which
is 0.14. This means that the total probability that the test will indicate
that the virus is present is 0.27 + 0.14, which is 0.41. We can now put
this probability onto the decision tree (Figure 8.11).
We can also use the probability tree in Figure 8.10(a) to calculate the
posterior probabilities of the virus being present or absent if the test gives
an indication of its presence. Using Bayes’ theorem, it can be seen that
these probabilities are 0.66 and 0.34, respectively. These probabilities
can also be added to the decision tree.
Prior probs Conditional probs Joint probs Posterior probs
0.27 0.27
= 0.66
0.41
0.3
Virus present
Virus absent
0.7
0.3
Virus present
Virus absent
0.7
0.9
Test indicates present
0.2
Test indicates present
p(test indicates virus present) = 0.41
(a)
0.14 0.14
= 0.34
0.41
Prior probs Conditional probs Joint probs Posterior probs
Test indicates absent
0.1
Test indicates absent
0.03 0.03
= 0.05
0.59
0.8
0.56 0.56
= 0.95
p(test indicates virus present) = 0.59 0.59
(b)
Figure 8.10 – (a) Revising the prior probabilities when the test indicates that the virus is
present; (b) revising the prior probabilities when the test indicates that the virus is absent