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216 Revising judgments in the light of new information
the chapter we will show how the potential benefits of information can
be evaluated so that a decision can be made as to whether it is worth
obtaining it in the first place.
Bayes’ theorem
In Bayes’ theorem an initial probability estimate is known as a prior
probability. Thus the marketing manager’s assessment that there was an
80% probability that sales of the calculator would reach break-even level
was a prior probability. When Bayes’ theorem is used to modify a prior
probability in the light of new information the result is known as a
posterior probability.
We will not put forward a mathematical proof of Bayes’ theorem here.
Instead, we will attempt to develop the idea intuitively and then show
how a probability tree can be used to revise prior probabilities. Let us
imagine that you are facing the following problem.
A batch of 1000 electronic components was produced last week at
your factory and it was found, after extensive and time-consuming
tests, that 30% of them were defective and 70% ‘OK’. Unfortunately, the
defective components were not separated from the others and all the
components were subsequently mixed together in a large box. You now
have to select a component from the box to meet an urgent order from
a customer. What is the prior probability that the component you select
is ‘OK’? Clearly, in the absence of other information, the only sensible
estimate is 0.7.
You then remember that it is possible to perform a ‘quick and dirty’
test on the component, though this test is not perfectly reliable. If the
component is ‘OK’ then there is only an 80% chance it will pass the
test and a 20% chance that it will wrongly fail. On the other hand, if
the component is defective then there is a 10% chance that the test will
wrongly indicate that it is ‘OK’ and a 90% chance that it will fail the
test. Figure 8.1 shows these possible outcomes in the form of a tree. Note
that because the test is better at giving a correct indication when the
component is defective we say that it is biased.
When you perform the quick test the component fails. How should
you revise your prior probability in the light of this result? Consider
Figure 8.1 again. Imagine each of the 1000 components we start off with,
traveling through one of the four routes of the tree. Seven hundred of
them will follow the ‘OK’ route. When tested, 20% of these components