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 apportionment 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 7s. 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 11s. 0d.—that is to say, this latter sum will procure him the same moral advantage as the condition of his funds after his stake. Whence we draw this conclusion: that the game is disadvantageous, even in the cases where the stake is equal to the product of the sum hoped for by the probability. We may, therefore, form an idea of the immorality of games in which the hoped-for sum is below this product.
Fig. 857.—The game of the needle.
Jacques Bernouilli has thus laid down the result of his investigations on the calculation of probabilities. An urn containing white and black balls is placed in front of the spectator, who draws out a ball, ascertaining its colour, and puts it back in the urn. After a sufficient number of draws, the total number of extracted balls divided by the total number of balls represents a fraction very near to that which has for a numerator the real number of white balls existing in the urn, and for the denominator the total number of balls. In other words, the ratios of the number either of extracted white balls, or the whole of the white balls to the total number, tend to become equal; that is, the probability derived from this experiment approaches indefinitely towards a certainty. The two fractions may differ from each other as little as possible, if we increase the number of draws. From this theorem we deduce several consequences.
1. The relations of natural effects are nearly constant when these effects are considered in a great number.
2. In a series of events indefinitely prolonged, the action of regular and constant causes affects that of irregular causes.
Applications.—The combinations presented by these games have been the subject of former researches regarding probabilities. We will complete our exposition with two more examples.
Two persons, A and B, of equal skill, play together on the understanding that whichever beats the other a certain number of times, shall be considered to have won the game, and shall carry off the stakes. After several throws the players agree to give up without finishing the game; and the point then to be settled, is in what manner the money is to be divided between them. This was one of the problems laid before Pascal by the Chevalier de Méré. The shares of the two players should be proportional to their respective probabilities of winning the game. These probabilities depend on the number of points which each player requires to reach the given number. A’s probabilities are determined by starting with the smallest numbers, and observing that the probability equals the unit, when player A does not lose a point. Thus, supposing A loses but one point, his chance is 1·2, 3·4, 7·8, etc., according as B misses one, two, or three points. Supposing A has missed two points, it will be found that his chance is as 1·4, 1·2, 11·6, etc., according as B has missed one, two, or three points, etc. Or we may suppose that A misses three points, and so on.
We should note, en passant, that this solution has been modified by Daniel Bernouilli, by the consideration of the respective fortune of the players, from which he deduces the idea of moral hope. This solution, famous in the history of science, bears the name of the Petersburgh problem, because it was made known for the first time in the “Memoires de l’Académie de Russie.”