its happening, at least t times, is the sum of the probabilities of its happening every time, of failing only once, twice, times; and the sum of these probabilities is ...... n t 444. Ex. 1. What is the probability of throwing an ace twice at least, in three trials, with a single die? In this case n = 3, t = 2, a = required is 445. 1+ 3 x 5 6 × 6 × Ex. 2. 6 1, b = 5; and the probability What is the probability that out of five individuals, of a given age, three at least will die in a given time? Let - m be the probability that any one of them will die in the given time (Art. 428); then we have given the probability of an event's happening in one instance, to find the probability of its happening three times, at least, in five instances. 1, b = m - 1, n = 5, t = 3; therefore the pro 1 + 5 (m − 1) + 10 (m − 1)2 m5 II. INVERSE PROBABILITIES. 446. AXIOM. When an event can proceed from one of a system of causes, the probabilities of these causes having produced the event are proportional to the numbers of ways in which they can severally produce the event. 447. COR. Hence the probabilities of the several causes having produced the event are proportional to the chances of the event happening on the assumption of their being severally existent. If P1, P2, P3, &c. be the probabilities of several causes from which an event may proceed; a1, a, a,, &c. the chances of that event happening on supposition of these causes severally existing; we have But as some one of the system of causes is known to be the true one, (p) is certainty, or 1, A Ex. An urn contains 3 balls which may be white or black. ball is drawn out and replaced three times, and in each case a white ball is drawn. What are the probabilities of the urn containing (1) Three white balls (2) Two white and one black: (3) One white and two black: (4) Three black? (1) Supposing the 1st state of the urn, the chance of the event happening which did happen is or that is, certainty. 3 3 3 27 X X 3 3 3 27 , The following Proposition involves the nature of both direct and inverse probabilities: 448. If a, a, a,, &c. be the chances of an observed event on supposition of each of the system of causes being the true one; a'', ag', a', &c. the probabilities of another proposed event on the same separate supposiΣ(aa') tions; the chance of this event happening = (a) For, as has been seen, a, a, a,, &c. are proportional to the chances given by the observed event of the separate causes existing. Hence the number of ways of the first cause existing and the second event happening from it is proportional to aa; and the number of ways of that event happening from some one of the causes is proportional to Σ (aa'). But since some one of the causes exists, (a) is in the same proportion to the number of ways in which one of the causes may exist, and the event either happen or fail from it. Hence the chance of the event happening (aa') Σ(α) Ex. 1. An urn contains two balls, but whether white or black, uncertain-we draw one ball, and find it is white. The ball is then replaced; what is the chance of next drawing a black one? Before the drawing takes place the two states or causes may be, (1) Two white, and (2) One white and one black. Now the chance of the observed event under (1) is (2) is 2 Again, the chance of the proposed event under (1) is 0, or 1, Ex. 2. Taking the case supposed in Ex. Art. 447, what is the chance of a white ball coming out at a fourth such drawing? Here before the drawing takes place the states or causes may be, (1) Three white, (2) Two white and one black, (3) One white and two black. Now the chance of the observed event under (1) is 1; Again, the chance of the proposed event under (1) is 1; 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 420, 422, 429, 441, 443, 447, and 448; 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. 1 6' The probability of throwing an ace with one die is and since 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. 437); 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. 449. 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 B left at the end of 1, 2, 3, &c. years; then is the value of the C D , A 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 &c. its value for 2, 3, &c. years respec 1 R (Art. 397); but it is only to be paid on condition that the * The more modern writers on this subject are Laplace, Galloway, and De Morgan. annuitant is alive at the end of the year, of which event the proba is AR therefore the Present Value of the conditional annuity (Art. 421); in the same manner, the Present Value of the C second year's annuity is ; the Present Value of the third year's D annuity is 1 B C AR3 + + D A R R2 R3 &c.; therefore the whole value required is + &c. to the end of the Tables.) 450. 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. + + + &c. A R R2 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 |