..the required fine = 2800 × {0.50505 - 0.35895} =2800 × 0.1461 = 409£, very nearly. SCHOLIUM. 418. The method of determining the present value of an annuity at simple interest, given in Art. 409, has been decried by several eminent Arithmeticians, and in its stead a solution of the question has been proposed upon the following principle; "If the present value of each payment be determined separately, the sum of these values must be the value of the whole annuity". Let be the value or price paid down for the annuity, a the yearly payment, n the number of years for which it is to be paid, r the interest of 1£ for one year. The present value of the first (Art. 397); the present value of the second pay α ment, or of a £ to be paid at the end of two ; and 1+2r a circumstance which the According to the former opponents seem not to have attended to. solution, no part of the interest of the price paid down is employed in paying the annuity, till the principal is exhausted. Let the annuity be always paid out of the principal a as long as it lasts, and afterwards out of the interest which has accrued; then x, x 2a, x-3a, &c. are the sums in hand, during the first, second, third, fourth, &c. years, the interest arising from which rx, rx − ra, rx 2ra, rx - 3ra, &c. that is, the whole in a, x terest, is nrx - {1 + 2 + 3 ... (n − 1)}× ra, or, nræ n-1 which, together with the principal x, is equal to the sum of all According to the other calculation, part of the interest, as it arises, is employed in paying the annuity, but not the whole. Thus, the first payment is made by a part of the principal, and the interest of that part, which together amount to the annuity; and the other payments are made in the same manner; this is, in effect, allowing interest upon that part of the whole interest which is incorporated with the principal. According to either calculation, the seller has the advantage, since the whole or a part of the interest will remain at his disposal till the last annuity is paid off. If the whole interest, as it arises, be incorporated with the principal, and employed in paying the annuity, compound interest is, in effect, allowed upon the whole. Let be the price paid for the annuity, n the number of years for which it is granted, and R1 together with its interest for one year. Then a in one year amounts to Rx, out of which the annuity being paid, Rx a is the sum in hand at the end of the first year; R2x is the amount of this sum at the end R2x Raa is the sum in hand at in the same manner, R" - R-1 a after paying the last annuity, which - Ra of the second year, therefore the end of the second year; Rn-2 a...- a is the sum left, ought to be nothing; hence CHANCES OR PROBABILITIES. 419. Chance, or Probability, has two meanings; the one a popular meaning, without any very distinct signification; the other a mathematical meaning, pointing out a real value existing in the circumstances. DEF. Most questions of probabilities will fall under one of two classes, called direct and inverse probabilities. A question of probability is termed direct, when, certain causes being given as existent, from which a certain event may proceed, the probability of that event happening is required. A question of probability is termed inverse, when, an event being given as existent, and proceeding from one of several causes, the probability of one proposed cause being the true one is required. Some more complex questions may partake of the nature of both kinds of probability. I. DIRECT PROBABILITIES. 1 n 420. If an event may take place in n different ways, and each of these be equally likely to happen, the probability that it will take place in a specified way is properly represented by certainty being represented by 1. Or, which is the same thing, if the value of certainty be 1, the value of the expectation that the event will happen in a specified way is 1 -· n For the sum of all the probabilities is certainty, or 1, because the event must take place in some one of the ways; and the probabilities are equal: therefore each of them is 421. COR. If the value of certainty be a, α 1 n the value of the expectation is But in the following Articles we suppose the n value of certainty to be 1. 422. If an event may happen in a ways, and fail in b ways, any of these being equally probable, the chance of its happening b a The chance of its happening must, from the nature of the supposition, be to the chance of its failing, as a b; therefore the chance of its happening: chance of its happening together with the chance of its failing a a+b. And the event must either happen or fail; consequently the chance of its happening together with the chance of its failing is certainty. Hence the chance of its happening certainty a a+b; or the chance of its happen Also, since the chance of its happening together with the chance of its failing is certainty, which is represented by 1, 1 423. Ex. 1. The probability of throwing an ace with a single 1 5 die, in one trial, is; the probability of not throwing an ace is 6; 2 the probability of throwing either an ace or a deuce is &c. 424. Ex. 2. If n balls, a, b, c, d, &c. be thrown promiscuously into a bag, and a person draw out one of them, the probability that it will be a is ; the probability that it will be either 425. Ex. 3. 1 n The same supposition being made, if two balls be drawn out, the probability that these will be a and b is For there are n. combinations of n things taken two and n- 1 two together (Art. 300); and each of these is equally likely to be taken; therefore the probability that a and b will be taken is 2 426. Ex. 4. If 6 white and 5 black balls be thrown promiscuously into a bag, and a person draw out one of them, the proba bility that this will be a white ball is 6 ; and the probability that 11 From the Bills of Mortality in different places Tables* have been constructed which shew how many persons, upon an average, out of a certain number born, are left at the end of each year, to the extremity of life. From such Tables the probability of the continuance of a life, of any proposed age, is known. 427. Ex. 1. To find the probability that an individual of a given age will live one year. * Some of these Tables will be found at the end of the Section. Let A be the number, in the Tables, of the given age, B the B number left at the end of the year; then is the probability that the individual will live one year; and he will die in that time (Art. 422). A In Dr. Halley's Tables, out of 586 of the age of 22, 579 arrive at the age of 23; hence the pro bability that an individual aged 22 will live one year is 1 1 84 579 586 or or nearly, is the 1 + probability that he will die in that time. 428. Ex. 2. To find the probability that an individual of a given age will live any number of years. Let A be the number in the Tables of the given age; B, C, D,...X, the number left at the end of 1, 2, 3,...x, years; then C A B is the probability that the individual will live 1 year; the A probability that he will live 2 years; and the probability that A-BA-CA-X A babilities that he will die in 1, 2, ≈ years, respectively. 429. If two events be independent of each other, and the probability that one will happen be and the probability that m and n being the numbers of ways in which the events can severally happen or fail. |