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The Elements of Probabilities

THE ELEMENTS OF PROBABILITIES. An understanding of the simpler elements of the science of probabilities is desirable before proceeding further; for, of course, life insurance, indeed all insurance, deals with probabilities.

The history of the science of probabilities is short. It was virtually unknown to the ancients, although traces of it are found in practice, as in the loans upon ships and cargoes, already referred to, and also in Ulpian's attempt to value life annuities. Traces are also found in the writ ings of Pythagoras and in the works of mathematicians of India.

Its first modern appearance is in 1663, in a pamphlet published in Italy by Cardan, and entitled "De ludo aim," i. e., "upon the game of dice." The next is in a series of letters written by Pascal, the great French savant, from 1654 to 1679, in which he discussed the chances at play. The next great work, and the first to put the subject into treatise form, was Jacob (or James) Bernoulli's "Ars Conjectandi," published in 1705. Then came Augustus de Moivre's "Doctrine of Chances," published in England in 1711, followed by Thomas Simpson's "Nature and Laws of Chance," published in 1740. The next great work was La Place's "Analytical Theory of Probabili ties," written and published by the French Government, under the patronage of the great Napoleon, in 1812. After this came De Morgan's "Probabilities," in England, in 1838, and Quetelet's "Letters on Probabilities," in French, in 1845. Whitworth's "Choice and Chance" is the best English text-book.

The fundamental discovery was that certainty could be represented by unity and a chance by a fraction, the value of which can be computed, the requisite data being pro vided.

Thus suppose it were known that one out of 36 persons known to have been on board a vessel had been lost, but not which one. Plainly it is an even chance, one with another, that a particular person was lost; and since there are 36 such equal chances, the chance as to any one per son is in the ratio, I to 36, which is Mo. The chance that he was not the person lost is in the ratio, 35 :36, which is Suppose that 6 were passengers and 3o were the crew; the chance that the person lost was a passenger is evidently 6 times as great as that he was a certain pas senger. Now let the news be received that the person was

a passenger, which then becomes a certainty. The chance that it was a particular passenger now becomes Us, i. e., one-sixth of unity; but the probability that it was a pas senger, we have found to be six times as great, i. e., 6 X 36 =-- i. That is, certainty is unity and all probalities are fractions.

That reasoning is clear and conclusive as applied to past events. Now let us assume that it is known that out of 36 trials, a thing will happen once and fail to happen 35 times. It is again evident that the chance of its hap pening is 1/46 and the chance of its failing to happen is This applies to the first trial.

Whether it would apply to further trials or not would depend upon whether its happening is exclusive of its recurrence or not. For instance, suppose it is drawing balls from a box which contains one black and 35 white balls. If a black ball is drawn the first time and is not replaced, nothing but white can be drawn afterward; if a white ball is drawn the first time, and is not replaced, the chance of a black ball the next time is not I :36 but :35.

But in either case, if the first ball is replaced, the chance remains the same, viz., I :36, or 148, for each trial.

Probabilities as to future events may be deduced in either of two ways, viz., by a priori reasoning or by knowl edge of what has happened before under conditions pre cisely similar. The infirmities of the human mind are such that reasoning from the nature of things is perilous unless supported by experiment or by observation. It is also in most matters very difficult to assure that past oc currences took place under conditions precisely similar to those which will hereafter obtain. Yet induction from the facts of experience affords in almost all things a safer basis for predictions of the future than does deduction from assumptions as to the nature of things. This is true, also, though the conditions be but approximately the same, and not exactly so.

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