CHANCES, doctrine of, in mixed mathe matics, a subject of great importance, es pecially as applied to the doctrine of life annuities, assurance, &c. in a great commercial country like this. The writers on this branch of science have been comparatively few. In our own language the principal treatises are, a large quarto by De Moivre, and a very small work by the celebrated Mr. Tho mas Simpson, in which, however, there are some problems never before attempt ed, or, at least, never before communicat ed to the public. In the year 1753, Mr. Dodson rendered this subject more acces sible to persons not far advanced in ana lytical studies, by publishing, in his se cond volume of the " Mathematical Re pository," a number of questions, with their several solutions, with an express reference to the doctrine of life annui ties. We shall give his first problem.
Suppose a round piece of metal, equal ly formed, having two opposite faces, one white, the other black, be thrown up, in order to see which of those faces will be uppermost after the metal has fallen to the ground, when, if the white face appears uppermost, a person is to be entitled to 5l. it is required to deter mine, before the event, what chance or probability that person has of receiving the 51. and what sum he may expect should be paid to him in consideration of his resigning his chance to another.
Solution. Since there is nothing in the form of the metal that can incline it to shew one face rather than the other, and since it must shew one, it will follow, that there is an equal chance for the appear ance of either face, or there is one chance out of two for the appearance of the white face, and consequently the proba bility of it may be expressed by the frac 1 tion if, therefore, any other person should be willing to purchase his chance, he must give for it the half of Si. or 21. 10s. This is one of the most simple ca ses : before, however, we proceed, it may be proper to give some definitions introductory to the doctrine.
Def. 1. The probability of an event is the ratio of the chance for its happening to all the chances for its happening or failing : thus, if out of six chances for its happening or failing, there were only two chances for its happening, the probabili ty in favour of such an event would be in the ratio of two to six; that is, it would be a fourth proportional to 6, 2, and 1, or 4.
For the same reason, as there are four chances for its failing, the probability that the event will not happen will be in the ratio of 4 to 6, or, in other words, it will be a fourth proportional to 6, 4, and 1, or Hence, if the fractions expressing the prbabilities of an event's both hap pening or failing be added together, they will always be found equal to unity. For let a be the number of chalices for the event's happening, and b the number of chances for its failing, the probability in the first case being and in the se a+ b cond case —' their sum will be = a+b a+b = 1. Having therefore determin ed the probability of any event's either happening or failing, the probability of the contrary will always be obtained by subtracting the fraction expressing such probability from unity.
Def. '2. The expectation of an event is the present value of any sum or thing, which depends either on the happening or on the failing of such an event. Thus, if the receipt of one guinea were to de pend on the throwing of any particular face on a die, the expectation of the per son entitled to receive it would be worth 3s. 6d.; for since there are six faces on a die, and only one of them can be thrown to entitle the person to receive his mo ney, the probability that such a face will be thrown being 4.. (according to Def. 1.) it follows, that the value of his interest before the trial is made, or, which is the same thing, that his expectation is equal to one-sixth of a guinea, or 38. 6d. Were his receiving the money to depend on his throwing either of two faces, his expecta tion would be equal to two-sixths of a guinea, or 7s. And, in general, supposing the present value of the money or thing to be received to be A, the probability of the event's happening to be denoted by a, and of its failing by b, the expectation A a b will be either expressed by---- or by A according as it depends either on a+b the event's happening, or on its failing.