There are three approaches to deriving probabilities: classical, relative frequency, and subjective. The classical and relative frequency approaches lead to objective probabilities, while the subjective approach can differ between people. The classical approach uses a formula involving outcomes and total possibilities. The relative frequency approach defines probability as the proportion of times an event occurs over many repetitions. Most decision problems require estimating unique event probabilities. Probability rules include addition for mutually exclusive and non-exclusive events, and the complement and independence rules.
1. 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
2. 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.
