MATHEMATICAL EXPECTATION.
A person who has a possibility of securing without cost a property of any sort places upon that possibility a value which depends upon that of the property and upon the probability of his securing it. In our subject a definite valuation is placed upon such a possibility.
This valuation is called its mathematical tion and is defined to be the product of the value of the property and the probability of getting it.
More generally, if there are several gains, with the respective probabilities, Pe, the mathematical expectation arising from these gains is + + ...
In this connection losses may be treated as negative gains. It is not denied that such an estimate involves the disregard of many elements, but unquestionably in the applications of this definition which are generally made such disregard is reasonable. To be more specific, it is extremely unlikely that a ..or man would regard a possible gain of $2,111, whose proba bility is k, as equivalent to the actual possession of $1,000. On the other hand such an estimate is made the basis of all insurance business and isjustified by the results obtained.
The connection between mathematical ex pectation and gambling is evident. If a person throws a die and is to receive $600 in case an ace appears, his expectation is $600 X It = $100. Unquestionably $100 would be regarded as a fair entrance-fee to one who could afford to repeat the trial a number of times. In this
connection the condition under which a game is said to be fair may be mentioned. Suppose that a player has to pay a stake b to enter a Fame in which his chance of winning a sum a is p. Then for a fair game pa, or his ex pectation after his entry must be equal to his stake.
The derivation from the definition of matical expectation of certain results which seemed to him to be impossible of acceptance led Daniel Bernoulli to develop a theory of moral expectation which has received some considerable attention. Bernoulli laid down the principle that to the possessor of a fortune a the moral value, v, of a small gain h is directly proportional to the amount of the increase, and inversely to this present fortune; or that — a (where k is a constant); and that if the probability of this gain is p, the moral expecta pkh tion arising from it is = _ The results a obtained from this estimate differ widely from those obtained under the preceding one. They do, however, become less widely divergent from the latter as a increases, and in fact as a in creases indefinitely the two sets of results tend to a perfect agreement.