EXPECTATION

, in the Doctrine of Chances, is applied to any contingent event, upon the happening of which some benefit &c is expected. This is capable of being reduced to the rules of computation: for a sum of money in Expectation when a particular event happens, has a determinate value before that event happens. Thus, if a person is to receive any sum, as 10l, when an event takes place which has an equal chance or probability of happening and failing, the value of the Expectation is half that sum or 5l.: but if there are 3 chances for failing, and only 1 for its happening, or one chance only in its favour out of all the 4 chances; then the probability of its happening is only 1 out of 4, or 1/4, and the value of the Expectation is but 1/4 of 10l. which is only 2l. 10s. or half the former sum. And in all cases, the value of the Expectation of any sum is found by multiplying that sum by the fraction expressing the probability of obtaining it. So the value of the Expectation on 100l. when there are 3 chances out of 5 for obtaining it, or when the probability of obtaining it is 3/5, is 3/5 of 100l. which is 60l. And if s be any sum expected on the happening of an event, h the chances for that event happening, and f the chances for its failing; then, there being h chances out of f + h for its happening, the probability will be h/(f + h), and the value of the expectation is h/(f + h) X s. See Simpson's or De Moivre's Doctrine of Chances.

Expectation of Life, in the Doctrine of Life Annuities, is the share, or number of years of life, which a person of a given age may, upon an equality of chance, expect to enjoy.

By the Expectation or share of life, says Mr. Simpson (Select Exercises pa. 273), is not here to be understood that particular period which a person hath an equal chance of surviving; this last being a different, and more simple consideration. The Expectation of a life, to put it in the most familiar light, may be taken as the number of years at which the purchase of an annuity, granted upon it, without discount of money, ought to be valued. Which number of years will differ more or less from the period above-mentioned, according to the different degrees of mortality to which the several stages of life are incident. Thus it is much more than an equal chance, according to the table of the probability of the duration of life (p. 254 ut supra), that an infant, just come into the world, arrives not to the age of 10 years; yet the Expectation or share of life due to it, upon an average, is near 20 years. The reason of which wide difference, is the great excess of the probability of mortality in the first tender years of life, above that respecting the more mature and stronger ages. Indeed if the numbers that die at every age were to be the same, the two quantities above specified would also be equal; but when the said numbers become continually less and less, the Expectation must of consequence be the greater of the two.

Mr. Simpson has given a table and rule for finding this Expectation, pa. 255 and 273 as above. Thus,

For Example, if it be required to find the Expectation or share of life, due to a person of 30 years old. Opposite the given age in the first column of the table, stands 23.6 in the second col. for the years in the Expectation sought.

See De Moivre's Doctrine of Chances applied to the Valuation of Annuities, p. 288; or Dr. Price's Observations on Reversionary Payments, p. 168, 364, 374, &c; or Philos. Trans. vol. 59, p. 89.