Probability statements

The mathematical theory of probability is concerned with the mathematical problems which arise in connexion with the manipulation of probability statements. We must therefore begin by taking a look at the way in which such statements can occur, and the relations between them.

The most primitive kind of probability statement first arose in the study of games of chance, and is of the following type:

(A) If a fair coin is tossed, the probability of its falling heads is 1/2;

The point of this statement is that it is impossible to predict which of the two outcomes, heads or tails, will occur. Moreover, there is symmetry between them (this being the burden of the adjective ‘fair’) so that, in some sense, neither is more likely than the other. Thus we assign to each outcome one half of the total probability, which by convention is 1.

Another statement of an entirely similar kind is

(B) If a die is thrown, the probability of a 5 being shown is 1/2;

Here there are six outcomes, supposed by symmetry to be equally likely, so that the probability ⅙is assigned to each From simple statements like (B) can be built up more complex ones, such as

(C) If a die is thrown, the probability of either a 2 or a 5 being shown is 1/6; + 1/6; = 1/3;

More generally, if E is an event made up of exactly r of the six possible outcomes, it is assigned a probability, denoted by P(E), of 1/6;r.