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220 Revising judgments in the light of new information
Events
Prior probabilities Conditional probabilities
0.75
At peak efficiency
Not at peak efficiency
0.25
New information
0.3
Temperature over 80°C
0.8
Temperature over 80°C
Joint probabilities
0.75 × 0.3 = 0.225
0.25 × 0.8 = 0.2
p(temperature over 80°C) = 0.425
Figure 8.3 – Applying Bayes’ theorem to the equipment operating problem
so the sum of the joint probabilities is: 0.225 + 0.2 = 0.425
and the required posterior probability is: 0.225/0.425 = 0.529
Another example
So far we have only applied Bayes’ theorem to situations where there are
just two possible events. The following example demonstrates that the
method of handling a problem with more than two events is essentially
the same.
A company’s sales manager estimates that there is a 0.2 probability
that sales in the coming year will be high, a 0.7 probability that they will
be medium and a 0.1 probability that they will be low. She then receives
a sales forecast from her assistant and the forecast suggests that sales
will be high. By examining the track record of the assistant’s forecasts
she is able to obtain the following probabilities:
p(high sales forecast given that the market will generate high sales) = 0.9
p(high sales forecast given that the market will generate only medium sales) = 0.6
p(high sales forecast given that the market will generate only low sales) = 0.3
What should be the sales manager’s revised estimates of the probability
of (a) high sales, (b) medium sales and (c) low sales?