PROBABILITY (Lat. probabilitas, from probabilis, probable). Expressions like the fol lowing are in common use: "It will probably rain to-day," "The chance of finding the article is very small," "He is more likely to succeed than to fail," "A is almost sure to be elected." These expressions all imply a lack of knowledge, an un certainty as to the actual condition of affairs. But they signify different degrees of uncertainty. The first and third are indefinite, the second and fourth are quite definite. In order to answer in mathematical terms the question, "What is the chance of an event happening?" it is neces sary to have some standard of measure or of com parison. Suppose we know only one of ten can didates on examination for a degree, and we hear that one passed. What is the chance or proba bility that our acquaintance is that one? lf, ac cording to our knowledge of the ease, one candi date is as likely to pass as any other, we may say that the ehanee of our acquaintance having passed is 1 to 10. lf, however, six of the candi dates are men. and our acquaintance is a man, and we hear that it is a man who passed, the chance is now 1 to 6. But if we hear the name of the successful candidate, this name corresponding to that of our acquaintance, and observe that the names on the list are all different, the chance is now 1 to 1, or it is a certainty. Certainty is called the unit of probability. It is the standard which all estimate alike. All other degrees of probability will be expressed as fractions of cer tainty. E.g., in the above ease of the candidate, on the first evidence the chance is 1:10, on the second evidence it is 1:6, on the third 1, or cer tainty.
If an event can occur in only one of a number of different ways, equally likely to occur, the probability of its happening at all is the sum of the several probabilities of its happening in the several ways. This proposition, the result of common experience, is generally accepted as axio matic. Thus a coin eon fall either head or tail, therefore the chance of its falling head is and of its falling tail is the sum of these chances being 1. This is as it should be, for the coin must certainly fall in order to produce bead' or tail. The probability of an event not happen ing is found by subtracting from unity the frac tion representing the probability that it will hap pen. E.g., if the chance of an event happening is
t, the chance of its not happening is 1 — r = Or, if A's chance of hitting a target is 4 and chance of hitting it is 4, the chance of both missing it is 1 —(4+ 4) = Two probabilities whose sum is unity are called complementary probabilities. E.g. the probability of drawing at one trial a white ball from a bag containing 2 white and 3 black balls is:4. The probability of drawing a black ball is 4. Their sum is 1, hence they are eompleHcntary probabilities. In gen eral, if an event can happen in a ways and fail in b ways, all of which are equally likely to oc cur, the probability of its happening is defined to a be a + and the probability of its failing is defined to bea + the two being complementary.
In this case, the odds in favor of the event are said to be a to b, and the odds against the event are said to be b to a. E.g. there are five ways of drawing one black ball from five black balls and three ways ot drawing a white one from three white balls. Hence the probability of drawing a black ball from the whole eight on the first trial is , and of not drawing a black ball, or, what is the same thing, of drawing a white one, is The odds in favor of drawing a black ball are 5 to 3. The odds against this are 3 to 5. Likewise. the odds in favor of drawing a white ball are 3 to 5, and the odds against it are 5 to 3.
If the probability of two independent events a' taking place are respectively and the probability that both will happen is aa' (a bfla'± The probability of both events fail bb' ing is (a+b)I O When fail is substituted for happen. bb' must be substituted for ae. Simi larly, the probability that the first event happens ab' and the second event fails is and („ + bpi+ b") the probability that the first event fails and the a'b second event happens is (a. E.g.. if p and p' are the respective probabilities that each of two events happens, then pp' is the proba bility that both happen. In like manner, if there are ant number of independent events. the proba bility that they will all happen is the product of their respective probabilities of happening.