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# The Elements of Probabilities

Another principle of the science of probabilities, only second in importance, then, is that, if out of a very large number of trials under conditions nearly alike, an event has been observed to take place a certain number of times, then, under like conditions, the probability that it will occur may be expressed by a fraction of which the number of times it happened is the numerator and the number of trials i. e., the number of times it happened plus the number of times it failed to happen is the denominator. And the chance that it will not occur may be expressed by a fraction with the same denominator, of which the num ber of times it did not happen is the numerator.

Thus, if out of 100,000 persons, at age ten, 676 have died in one year, we consider that the chance that a person, aged 1o, will die in one year is , and the chance that he will not die is 1909400, while the chance that he will die, plus the chance that he will not die is 67=24 100000 1, or certainty.

io In In the application of this important principle lies the pos sibility of computing in advance, within reasonable limits of error, the cost of insurance. The value of the risk or probability is estimated as closely as possible by means of averages drawn from past experience. The utmost care is, of course, necessary to ascertain that the condi tions of the past experience, from which the averages are derived, and of the future as to which insurance is to be given, are alike in all essential regards.

In life insurance alone are compound probabilities sometimes employed; as, for instance, in an insurance payable upon the failure of the first of two or more lives or an annuity continuing during the life of the last sur vivor of two or more lives.

The chance that two events will happen the first neither including nor excluding the second is not the sum of the chances that they will severally happen, but the product of those chances.

Thus suppose the chance that one thing will happen is lo that a second will happen is o0and that a third will happen is 1U Then out of i,000,000 persons, as to each of whom these chances are equally valid, the first thing will happen to i,000; and out of this I,000, to each of whom the second chance is equally valid, the second thing will happen to io; and out of these 1o, to each of whom the third chance is equally valid, the third thing will happen to i. Therefore, all these things will happen

to but one out of the original i,000,000; and the chance that all will happen to a particular one of them is 00000 This is equal to the several chances multiplied together, 1 l000 loo to 1000000• In a similar manner it may be proved generally that where events are not mutually inclusive or exclusive, the probability that two or more will happen is the product of the probability that each will happen separately.

So, the chance of incurring a claim within a year on life insurance on two joint lives is the chance that one will die plus the chance that the other will die, less the chance that both will die, i e., less the product of the two chances, for the amount of the policy is paid but once. Or it may be given as certainty, which is I, less the chances that both will survive the year, which is the product of the chance that one will survive and the chance that the other will survive; for, if both do not survive the year, one at least must have died within it.

An annuity payable during the life of the survivor of two lives, for instance, will yield a payment at the end of the year if either life survive; that is, if both lives survive, or if the first life survive and the other expire, or if the first life expire and the second survive. The total probability, then, is the sum of these three separate probabilities.

No attempt will be made in this volume to develop these principles into mathematical formulas; but the reader will not fail to find the foregoing principles both interest ing and useful if he will give their statement in non mathematical language his careful attention.

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