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Bayes’ theorem 219
probability that the component is ‘OK’ we select the joint probability
which emanates from the ‘component is OK’ part of the tree, i.e. 0.14.
Thus the posterior probability is 0.14/0.41, which is, of course, 0.341.
The steps in the process which we have just applied are summarized
below:
(1) Construct a tree with branches representing all the possible events
which can occur and write the prior probabilities for these events on
the branches.
(2) Extend the tree by attaching to each branch a new branch which
represents the new information which you have obtained. On each
branch write the conditional probability of obtaining this information
given the circumstance represented by the preceding branch.
(3) Obtain the joint probabilities by multiplying each prior probability
by the conditional probability which follows it on the tree.
(4) Sum the joint probabilities.
(5) Divide the ‘appropriate’ joint probability by the sum of the joint
probabilities to obtain the required posterior probability.
To see if you can apply Bayes’ theorem, you may find it useful to attempt
the following problem before proceeding. A worked solution follows
the question.
Example
An engineer makes a cursory inspection of a piece of equipment and
estimates that there is a 75% chance that it is running at peak efficiency.
He then receives a report that the operating temperature of the machine is
exceeding 80 ◦ C. Past records of operating performance suggest that there
is only a 0.3 probability of this temperature being exceeded when the
machine is working at peak efficiency. The probability of the temperature
being exceeded if the machine is not working at peak efficiency is 0.8.
What should be the engineer’s revised probability that the machine is
operating at peak efficiency?
Answer
The probability tree for this problem is shown in Figure 8.3. It can be
seen that the joint probabilities are:
p(at peak efficiency and exceeds 80 ◦ C) = 0.75 × 0.3 = 0.225
p(not at peak efficiency and exceeds 80 ◦ C) = 0.25 × 0.8 = 0.2