PROBABILITY, Theory of, is that branch of mathematics which deals with the determina tion of the degree of belief which, in the ab sence of full information, should be given to certain classes of statements, or to the past or future occurrences of certain events.
In regard to the great majority of events there is in the mind of every one a state of un certainty. Under differing circumstances one shows this uncertainty by saying that it is pos sible or that it is, or is not, probable that some event did or will happen. Ordinarily no attempt is made to establish an exact measure of the probability of an event, such attempt being clearly foredoomed to failure. Not infrequently, however, the first step toward such a measure ment is taken by the making of a rough com parison of the probabilities of two events, the conclusion resulting that one of them is more probable than the other.
Equally Probable For certain sorts of events it is possible to proceed farther. Thus, as a next step, it is quite common to find that two events are regarded as equally prob able. For instance, a coin being tossed, one says that head and tail are equally likely to ap pear; or, a properly made die being thrown, one estimates that any one of the six faces is as likely to appear as any other. Of course these estimates are made in spite of the belief that, starting from a given initial position, the body moves completely subject to the laws of dynam ics, and that from certain influences but one result can follow. The fact is that the observer is so ignorant of the forces applied to the body that his judgment is formed independently of them. He perceives that from the nature of
the body only a certain number of events are possible, and he finds no reason for concluding that one event rather than another will occur.
Probability of an Event.— Suppose now that five events are known to be equally prob able and that one of them will and only one can happen. Suppose also that if and only if one or another of the first three occurs, a further event E will occur. Then it is commonly said that the odds are 3 to 2 in favor of the event E; and since there are five equally probable events, of which exactly three are concurrent with E, it is said that the mathematical probability of E is f. In general, when a specified event E is governed by n equally probable events, of which one will and only one can happen, and when, of nt of these, each is decisively favorable to E all others being decisively unfavorable, the mathematical probability of E is defined to be In the extreme cases in which none and all of the events, respectively, are favorable to E, the probabilities are said to be 0 and 1, though in each case there is certainty regarding the oc currence of E.
Assume now that, of n equally probable and mutually exclusive governing events, tn, m,, ..., ma are respectively favorable to the further events E1, which are also mutually exclusive. Then ihe probabilities of El, E,, MI ' Ea are respectively p, = — tn, ..... Ph= n — . If now nil m, ma = to, one has