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Exercises 237
the reliability of the new information. Of course, receiving information is
often a sequential process. Your prior probability will reflect the information
you have received up to the point in time when you make your initial
probability assessment. As each new instalment of information arrives,
you may continue to revise your probability. The posterior probability
you had at the end of last week may be treated as the prior probability
this week, and be revised in the light of this week’s information.
We also looked at how the value of new information can be assessed.
The expected value of perfect information was shown to be a useful
measure of the maximum amount that it would be worth paying for
information. Calculating the expected value of imperfect information
was seen to be a more involved process, because the decision maker
also has to judge the reliability of the information. Because of this, we
stressed the importance of sensitivity analysis, which allows the decision
maker to study the effect of changes in these assessments.
Exercises
(1) The sales of a magazine vary randomly: in 70% of weeks they are
classified as being ‘high’ while in 30% of weeks they are classified
as ‘low’.
(i) Write down prior probabilities of high and low sales in the
coming week .........
(ii) You are now given the sales figures for Monday and these show
low sales. In the past:
In weeks when sales turned out to be high, Monday had low
sales on only 20% of occasions;
In weeks when sales turned out to be low, Monday had low
sales on 60% of occasions.
Revise your prior probabilities in the light of Monday’s sales
figures.
(2) In January a sales manager estimates that there is only a ‘30%
chance’ that the sales of a new product will exceed one million units
in the coming year. However, he is then handed the results of a
sales forecast. This suggests that the sales will exceed one million
units. The probability that this indication will be given when sales
will exceed a million units is 0.8. However, the probability that
the forecast will give this indication when sales will not exceed a