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292 Methods for eliciting probabilities
Defective
weld
Figure 10.5 – A fault tree
Pipeline
fracture
or
0.001
Safety
valve
failure
0.001 + 0.002 − (0.001 × 0.002)
= 0.002998
Excess
pressure
and
0.02
0.02 × 0.1 = 0.002
Regulator
failure
The decision maker can now assess probabilities for the lowest-level
events (these probabilities are shown on the tree) and then work up
the tree until he eventually obtains the probability of the pipeline
fracturing. Since the safety valve and regulator failures are considered
to be independent their probabilities can be multiplied to obtain the
probability of excess pressure. This probability can then be added to
the probability of a weld being defective to obtain the probability of the
pipe fracture occurring. (Note that since excess pressure and a defective
weld are not mutually exclusive events the very low probability of them
both occurring has been subtracted from the sum – see Chapter 4.) Of
course, in practice most fault trees will be more extensive than this one.
Indeed, the decision maker in this problem may wish to extend the tree
downwards to identify the possible causes of safety valve or regulator
failure in order to make a better assessment of these probabilities.
Using a log-odds scale
Because people generally have problems in distinguishing between
probabilities like 0.001 and 0.0001 some analysts prefer to use what is
known as a log-odds scale to elicit the probabilities of rare events. You
will recall that odds represent the probability that an event will occur
divided by the probability that it will not. By converting probabilities
to odds and then taking the logarithms of the results we arrive at a
scale like that shown in Figure 10.6 (this figure shows the scale only
0.1