SCHOLIUM. Much more might be said on a subject so extensive as the doctrine of Chances; the Learner will however find the principal grounds of calculation in Articles 425, 427, 434, 446, 448, 452, and 453; and if he wish for farther information, he may consult De Moivre's work on this subject*. It may not be improper to caution him against applying principles which, on the first view, may appear self-evident; as there is no subject in which he will be so likely to mistake as in the calculation of probabilities. A single instance will shew the danger of forming a hasty judgment, even in the most simple case. The probability of throwing an ace with one die is and since 1 6 there is an equal probability of throwing an ace in the second trial, it might be supposed that the probability of throwing an ace in two This is not a just conclusion (Art. 442); for, it would follow, by the same mode of reasoning, that in six trials a person could not fail to throw an ace. The error, which is not easily seen, arises from a tacit supposition that there must necessarily be a second trial, which is not the case if an ace be thrown in the first. LIFE ANNUITIES. 454. To find the Present Value of an annuity of £1, to be continued during the life of an individual of a given age, allowing compound interest for the money. Let R be the amount of £1 in one year; A the number of persons in the Tables of the given age; B, C, D, &c. the number left at the end of 1, 2, 3, &c. years; then is the value of the B A &c. its value for 2, 3, &c. years respec C D life for one year, A' A tively; and the series must be continued to the end of the Tables. Now the Present Value of £1, to be paid at the end of one year, is 1 R (Art. 408); but it is only to be paid on condition that the * The more modern writers on this subject are Laplace, Galloway, and De Morgan. ED. annuitant is alive at the end of the year, of which event the proba bility is B is AR B A ; therefore the Present Value of the conditional annuity (Art. 426); in the same manner, the Present Value of the 1 B C D &c.; therefore the whole value required is X + +&c. to the end of the Tables.) A R R1 455. R$ 3 De Moivre supposes, that out of eighty-six persons born, one dies every year, till they are all extinct. This supposition is sufficiently exact, if our calculations be made for any age neither very young nor very old, as will appear from an inspection of the Tables; and, on this supposition, the 1 B C D sum of the series X + + +&c.) may be found. A RR R3 Let n be the number of years which any individual wants of 86; then will n be the number of persons living, of that age, out of which one dies every year; and will be the probabilities of his living 1, 2, 3, &c. years; hence, the Present Value of an annuity of £1, to be paid during his life, is 456. COR. 1. This expression for the sum is the same with Let P be the Present Value of an annuity of £1, to continue 457. COR. 2. The Present Value of the annuity to continue for ever, from the death of the proposed individual, is RP n (R − 1) For the whole Present Value of the annuity, to continue for (Art. 420); and if from this its value for the life of the individual be taken, the remainder is the Present RP Value of the annuity, to continue for ever, from the time of his death. 458. To find the Present Value of an annuity of £1, to be paid as long as two specified individuals are both living. Find, by Art. 444, the probability that they will both be alive at the expiration of 1, 2, 3, &c. years, to the end of the Tables; call these probabilities a, b, c, &c. and R the amount of £1, in one α b с year; then + + + &c. is the Present Value of the annuity R R2 R3 required. (See Art. 454.) 459. To find the Present Value of an annuity of £1, to be paid as long as either of two specified individuals is living. Find, by Art. 445, the probability that they will not both be extinct in 1, 2, 3, &c. years, to the end of the Tables, and call these probabilities A, B, C, &c.; then the Present Value of the annuity is 460. COR. If the annuity be M£, the Present Value is M times as great as in the former case, or 461. These are the mathematical principles on which the values of annuites for lives are calculated, and the reasoning may easily be applied to every proposed case. But, in practice, these calculations, as they require the combination of every year of each life with the corresponding years of every other life concerned in the question, will be found extremely laborious, and other methods must be adopted when expedition is required. Writers on this subject are De Moivre, Maseres, Simpson, Price, Morgan, and Waring. Other writers on the subject are Milne, Baily, and De Morgan, of which the last mentioned is now most accessible. 462. To find the Present Value of the next presentation to a Living. Let i£ be the average annual net income of the living; c£ the cost of a curate, that is, the money value of the work to be done; and a£ the unavoidable expences of the admission of a new incumbent. Then the Present Value of the next presentation will obviously be the present value of an annuity of (i-c)£, to commence at the death of the present incumbent, and to continue during a life then 24 years of age, deducting the present value of a£ payable on admission. Let n be the number of years which the Tables give to the present incumbent, p the number for a person 24 years of age; then the Present Value required will be that of an annuity of (i-c)£ to commence at the expiration of n years, and to continue p years, deducting the Present Value of c£ to be paid after n years*, = i-c R-1 COR. The Present Value of an Advowson, or perpetual nomination to a living, will be that of an Annuity of (i-c)£ commencing after n years, and continuing for ever, deducting the present value of a£, to be paid at the end of n years, and also of the same sum to be paid at intervals of p years for ever afterwards, * Of course this is on the supposition, that the laws, which permit such traffic in spiritual cures, remain in force, and that the values of i, c, and a remain unaltered, for n+p years at least. |