As well as the subjective, objective, and frequentist approaches to probability, there are other standpoints. For example, should one always insist on associating a probability with a number? Might it be enough to say that one probability was greater, or one degree of belief was more intense, than another? And should we necessarily offer an initial set of axioms – self-evident truths – on which to erect a theory?
Many distinguished writers have felt it useful to have two separate approaches, one for degrees of belief and one for objective probabilities. Both would have the same rules of logic, free from contradictions, but how values of probabilities were arrived at, and how they are interpreted, could differ. Any theory should be consistent with the classical view, based on repeatable experiments with equally likely outcomes. So we will focus on that case, seeking any rules that the notion of probability must obey.
The Addition Law
Deal one card from a well-shuffled pack. We take all cards as equally likely, so the probability of any event, such as obtaining a Club, or a Spade, or an Ace is found by calculating the proportion of all possible outcomes that lead to those events. How might we find the probability that either of two such events occur?If those events have no outcomes in common, we say that they are mutually exclusive , or disjoint . The events ‘Get a Spade’ and ‘Get a Club’ are disjoint, but the events ‘Get a Spade’ and ‘Get an Ace’ are not disjoint, as the Ace of Spades belongs to both. When two events are mutually exclusive, then the total number of outcomes that lead to either event is just the sum of the numbers for each event separately, so we have a simple result: whenever two events are mutually exclusive ,
the probability that at least one occurs is the sum of their individual probabilities.
This is the Addition Law . It plainly holds in all experiments where we would take the classical view: using the balls in a bag analogy, it is the same as saying that the total number of balls that are either Red or Blue is the sum of the number of Red balls and the number of Blue balls. And in any repeatable experiment, such as rolling dice,