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

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.