* MATHEMATICAL HOPE." 727 



etnd another expected event, the second probability divided by the first, will 

 be the probability of the expected event, inferred from the observed event. 



The probability of events serves to determine the hope or fear of 

 persons interested in their existence. The word hope here expresses the 

 advantage which someone expects in suppositions which are only probable. 

 This advantage in the theory of chances is the product of the hoped-for 

 sum by the probability of obtaining it ; it is the partial sum which should arise 

 when one does not wish to run the risks of the event, supposing that the appor- 

 tionment corresponds to the probabilities. This apportionment is only 

 equitable when we abstract from it all foreign circumstances; because an equal 

 degree of probability gives an equal title to the hoped-for sum. This advantage 

 is called mathematical hope. Nevertheless, the rigorous application of this 

 principle may lead to an inadmissible consequence. Let us see what Laplace 

 says. Paul plays at heads and tails, on the understanding that he receives 

 two shillings if he succeeds at the first throw, four shillings if he succeeds at 

 the second, eight at the third, and so on. His stake on the game, according 

 to calculation, must be equal to the number of throws ; so that if the game 

 continues indefinitely, the stake also continues indefinitely. Yet, no reasonable 

 man would venture on this game even a moderate sum, 2 for example. 

 Whence, therefore, comes this difference between the result of the calculation, 

 and the indication of common-sense ? We soon perceive that it proceeds 

 from the fact, that the moral advantage which a benefit procures for us is 

 not proportional to this advantage, and that it depends on a thousand 

 circumstances, often very difficult to define, but the chief and most important 

 of which is chance. In fact, it is evident that a shilling has much greater 

 value for one who has but a hundred than for a millionaire. We must, 

 therefore, distinguish in the, hoped-for good between its absolute and its 

 relative value ; the latter regulates itself according to the motives which 

 cause it to be desired, while the former is independent. In the absence of a 

 general principle to appreciate this relative value, we give a suggestion of 

 Daniel Bernouilli which has been generally admitted. 



The relative value of an extremely small sum is equal to its absolute 

 value, divided by the total advantage of the interested person. On applying 

 the calculus to this principle, it will be found that the moral hope, the growth 

 of chance due to expectations, coincides with the mathematical hope, when 

 chance, considered as a unit, becomes infinite in proportion to the variations 

 it receives from expectations. But when these variations are a sensible 

 portion of the unit, the two hopes may differ very greatly from each other. 

 In the example cited, this rule leads to results conformable to the indications 

 of common-sense. We find, in point of fact, that if Paul's fortune amounts 

 only to 8, he cannot reasonably stake more than ?s. on the game. At 

 the most equal game, the loss is always relatively greater than the gain. 

 Supposing, for example, that a person possessing a sum of 4, stakes 2 on 

 a game of heads or tails, his money after placing his stake will be morally 

 reduced to 3 1 1 s. od. that is to say, this latter sum will procure him tJie 



