THE THEORY OF PROBABILITY 185 69. Mathematical expectation. The name mathematical expec- tation has been given to the product of a sum whose payment depends upon the happening of some contingent event, multi- plied by the probability that the event will happen. For example, if a stake of $6000 were offered, to be paid if the throw of a die is 1, the mathematical expectation of a player is $1000. This does not mean, of course, that a man having $1000 could afford to risk it on the throw of a single die, but rather that, by paying $1000 for each throw, in a very large number of throws the player would come out approximately even in the long run, in a perfectly fair game. A second example may serve to make the matter clearer. Suppose that 100,000 men, all aged forty years, agree to make up a fund of such an amount that at the end of a year each survivor may receive one dollar. What is the mathematical expectation of each participant ? It has been found by the American life-insurance companies that, of every 100,000 men aged forty years, approximately 979 will die within a year. The probability that any one of these 99 021 men will live one year is therefore . = .99021. If we -Lw^Vw neglect interest, the mathematical expectation of any one of the number is, then, $0.99021. The matter will be still clearer if we examine it from another point of view. If 979 die within a year, there will be 99,021 survivors, each to receive a dollar, so that $99,021 must be contributed. The amount that each man must contribute is therefore Aqq »oi -i?»»,U^i /,„ nn.-in-.
100,000
= $0.99021.
While the notion of mathematical expectation had its origin at the gaming table, it has been made the basis of one of the great economic developments of modem times, viz. the develop- ment of the business of life insurance. The example just given shows that if a man forty years old wishes to provide $10,000 for his estate in case his death should occur within one year, he would have to pay at least $97.90 to the insurance company