Probability

PROBABILITY (in Logic) is commonly contrasted with certainty. Some of our beliefs or judgments are entertained with certainty, others there are of which we are not so sure. The de grees of confidence with which the logic of probability is concerned are those which are correlated with different kinds of objective evidence, different degrees of objective cogency, not with the con fidence which depends on mere feeling, or arises we know not how. In other words, we are concerned with degrees of rational belief or confidence. Again, the degrees of confidence attach to the beliefs or the judgments, or the propositions expressing the beliefs or judgments, so that strictly speaking the probability refers to the judgments or propositions, not to the things or events to which these refer. Things just are, and events just happen—there is no certainty or probability in them. Only our judgments about them can be more or less probable.

Cases of calculable probability are of two main types, namely, those which can be calculated a priori, or deductively, and those which can only be calculated a posteriori, or inductively. The a priori type is that in which the calculation can be made by reasoning deductively from the nature of the case, and without reference to actual observations of the kind of events under con sideration. The a posteriori type consists of those cases in which the calculation can be made only with the aid of previous obser vations of similar events.

The A Priori Calculation of Probability.

In order to be able to calculate the probability of an event deductively from the nature of the case, the following conditions must be satisfied : (a) We must know the total number of mutually exclusive possi bilities, one or other of which must be realized. (b) These possi bilities must be equally likely. And (c) we must know how many of these possibilities are favourable to the event contemplated, that is, in how many of them it will be realized. The probability of the event in question is then expressed by means of a fraction in which the numerator gives the number of possibilities favour able to the event, and the denominator states the total number of equally likely possibilities. Clearly, the greater the total number of possibilities, the smaller is the probability (or chance) of any special one of them being realized; that is, if p represents the probability of an event and t the total number of equally likely events of which it is one, then p and t will vary inversely. For

example, the chance of throwing head when tossing a coin is greater than that of throwing face six when throwing a die, be cause the former result is one of two possibilities, whereas the latter is only one of six. On the other hand, the greater the number of possibilities that are favourable, the greater is the probability. For instance, when throwing a die the probability of getting an even number is greater than that of throwing face six in particular, because there are three even numbers and only one six on a die. In this way, if the number of favourable possi bilities be represented by f, we get the following general formula for probability : p= f/t.

A word may also be said about "odds" and "chances," terms which are more in popular favour than is the term "probability." The term "chances" is used sometimes for "probability" and sometimes for "odds." By "odds" is meant the ratio of favourable to unfavourable possibilities. By the odds against an event is meant the ratio of the unfavourable to the favourable possibilities.

In the case of simple events, like those of throwing a die or tossing a coin, there is no difficulty whatever in determining the values of favourable possibilities and of the total number of equally likely events and therefore of probability. With complex events (that is, those in which two or more separate events can be distinguished) care has to be exercised. The total number of pos sibilities in such cases is not the sum of the possibilities of the separate events, but their multiple, e.g., if a die is thrown twice the total number of possibilities is not 6+6, but 6X 6 or 36, because for each possible result of the first throw there are six possibilities with the second throw. Now, the probability of a complex event may, according to circumstances, be either greater or less than the probability of the separate component events. The probability is less if the complex event contemplated is one in which certain component events must occur in a certain order, say i followed by 6 in two throws of a die, for either of the component events might happen without the other also happening, and then one of the component events would be there but not the compound event. In such cases the probability is obtained by multiplying the fractions expressing the separate probabilities of the several component events. On the calculations of probability see PROBABILITY AND ERROR.