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Bayes’ theorem 217
1000
components
70%
Component is OK
Component is defective
30%
700 components
80% pass test
20% fail test
300 components
10% pass test
90% fail test
Figure 8.1 – Tree diagram for the components problem
140 components
270 components
410 components fail test
(i.e. 140) would be expected to wrongly fail the test. Similarly, 300
components will follow the ‘defective’ route and of these 90% (i.e. 270)
would be expected to fail the test. In total, we would expect 410 (i.e.
140 + 270) components to fail the test. Now the component you selected
is one of these 410 components. Of these, only 140 are ‘OK’, so your
posterior probability that the component is ‘OK’ should be 140/410,
which is 0.341, i.e.
p(component OK|failed test) = 140/410 = 0.341
Note that the test result, though it is not perfectly reliable, has caused
you to revise your probability of the component being ‘OK’ from 0.7
down to 0.341.
Obviously, we will not wish to work from first principles for every
problem we come across, and we therefore need to formalize the application
of Bayes’ theorem. The approach which we will adopt, which is
based on probability trees, is very similar to the method we have just
applied except that we will think solely in terms of probabilities rather
than numbers of components.
Figure 8.2 shows the probability tree for the components problem.
Note that events for which we have prior probabilities are shown on
the branches on the left of the tree. The branches to the right represent
the new information and the conditional probabilities of obtaining this
information under different circumstances. For example, the probability