Download - Probability decision making
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There are three different approaches to deriving probabilities: the classical approach,
the relative frequency approach and the subjective approach. The first two methods lead to
what are often referred to as objective probabilities because, if they have access to the same
information, different people using either of these approaches should arrive at exactly the
same probabilities. In contrast, if the subjective approach is adopted it is likely that people
will differ in the probabilities which they put forward. The classical approach to probability
involves the application of the following formula:
The probability of an event occurring = Number of outcomes which represent the
occurrence of the event / Total number of possible outcomes
In the relative frequency approach the probability of an event occurring is regarded as
the proportion of times that the event occurs in the long run if stable conditions apply. This
probability can be estimated by repeating an experiment a large number of times or by
gathering relevant data and determining the frequency with which the event of interest has
occurred in the past. Most of the decision problems which we will consider in this book will
require us to estimate the probability of unique events occurring (example events which only
occur once). Two events are mutually exclusive (or disjoint) if the occurrence of one of the
events precludes the simultaneous occurrence of the other. 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). If the events are
mutually exclusive then the addition rule is:
p(A or B) = p(A) + p(B) (where A and B are the events)
If the events are not mutually exclusive we should apply the addition rule as follows:
p(A or B) = p(A) + p(B) − p(A and B)
If A is an event then the event ‘A does not occur’ is said to be the complement of A. The
complement of event A can be written as A (pronounced ‘A bar’). Since it is certain that
either the event or its complement must occur their probabilities always sum to one. This
leads to the useful expression:
p(A) = 1 − p(A)
Two events, A and B, are said to be independent if the probability of event A
occurring is unaffected by the occurrence or non-occurrence of event B. For example, the
probability of a randomly selected husband belonging to blood group O will presumably be
unaffected by the fact that his wife is blood group O(unless like blood groups attract or
repel!). Similarly, the probability of very high temperatures occurring in England next August
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will not be affected by whether or not planning permission is granted next week for the
construction of a new swimming pool at a seaside resort. If two events, A and B, are
independent then clearly:
p(A|B) = p(A)
because the fact that B has occurred does not change the probability of A occurring. In other
words, the conditional probability is the same as the marginal probability.
Probability assessments are a key element of decision models when a decision maker
faces risk andsubjective, but must still conform to the underlying axioms of probability
theory. Uncertainty, In most practical problems the probabilities used will be. Probability
calculus is designed to show you what your judgments should look like if somebody accept
its axioms and think rationally. The correct application of the rules and concepts which we
have introduced in this chapter requires both practice and clarity of thought. You are
therefore urged to attempt the following exercises before reading further.