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The effect of new information 221
Events
New information
Prior probabilities Conditional probabilities
0.2
Medium sales
0.7
High sales
Low sales
0.1
Forecast suggests high sales
0.9
Forecast suggests high sales
0.6
Forecast suggests high sales
0.3
Joint probabilities
0.2 × 0.9 = 0.18
0.7 × 0.6 = 0.42
0.1 × 0.3 = 0.03
p(forecast suggests high sales) = 0.63
Figure 8.4 – Applying Bayes’ theorem to the sales manager’s problem
The tree for this problem is shown in Figure 8.4. The joint probabilities
are:
p(high sales occur and high sales forecast) = 0.2 × 0.9 = 0.18
p(medium sales occur and high sales forecast) = 0.7 × 0.6 = 0.42
p(low sales occur and high sales forecast) = 0.1 × 0.3 = 0.03
so the sum of the joint probabilities is 0.63
which means that we obtain the following posterior probabilities:
p(high sales) = 0.18/0.63 = 0.2857
p(medium sales) = 0.42/0.63 = 0.6667
p(low sales) = 0.03/0.63 = 0.0476
The effect of new information on the revision
of probability judgments
It is interesting to explore the relative influence which prior probabilities
and new information have on the resulting posterior probabilities.
Suppose that a geologist is involved in a search for new sources of
natural gas in southern England. In one particular location he is asked
to estimate, on the basis of a preliminary survey, the probability that gas
will be found in that location. Having made his estimate, he will receive
new information from a test drilling.