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76 Introduction to probability
we will introduce a number of elicitation methods which are designed
to help decision makers to make judgments about probabilities. At this
stage, however, it is worth pointing out that such judgments rarely
need to be exact. As we shall see, sensitivity analysis often reveals that
quite major changes in the probabilities are required before it becomes
apparent that the decision maker should switch from one course of
action to another.
Having looked at the three approaches to probability, we now need
to consider the concepts and rules which are used in probability calculations.
These calculations apply equally well to classical, relative
frequency or subjective probabilities.
Mutually exclusive and exhaustive events
Two events are mutually exclusive (or disjoint) if the occurrence of one
of the events precludes the simultaneous occurrence of the other. For
example, if the sales of a product in the USA next year exceed 10 000
units they cannot also be less than 10 000 units. Similarly, if a quality
control inspection of a new TV set reveals that it is in perfect working
order it cannot simultaneously be defective. However, the events of
‘dollar rises against the yen tomorrow’ and ‘the Dow-Jones index falls
tomorrow’ are not mutually exclusive: there is clearly a possibility that
both events can occur together. If you make a list of the events which
can occur when you adopt a particular course of action then this list is
said to be exhaustive if your list includes every possible event.
The addition rule
In some problems we need to calculate the probability that either one
event or another event will occur (if A and B are the two events, you
may see ‘A or B’ referred to as the ‘union’ of A and B). For example, we
may need to calculate the probability that a new product development
will take either 3 or 4 years, or the probability that a construction project
will be delayed by either bad weather or a strike. In these cases the
addition rule can be used to calculate the required probability but,
before applying the rule, it is essential to establish whether or not the
two events are mutually exclusive.