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25.) Cor. 2. The probability of the event's happening at least 1 imes in n trials is
n-1 a" + naa-sbtn. ar-18 ..., to n-6+1 terms
(a+b). For if it happen every time, or fail only once, twice,....nak it happens t times; therefore the whole probability of its happening at least t times, is the sum of the probabilities of its happening every time, of failing only once, twice,....n-t times ; and the sum of these probabilities is
n-1 q" + nan-+ n. aa-202.... to net+terms.
(a+b)" (26.) Er. 1. What is the probability of throwing an ace, twice, at le in three trials, with a single die?
In this case, n=3,1=2, a=), b=5; and the probability required
6.6 6.216 27
(27.) Er. 2. What is the probability that out of 5 individuals, of a given age, three, at least, will die in a given time.
1 Let be the probability that any one of them will die in the given time (Art. 10)., then we have given the probability of ao event's happening in one instance, to find the probability of its happening three times in five instances. In this case, a=1, l=m-1, n=5, t=3, therefore the probability
1 +5(.m-1)+10.(m-1) required is
(27.) 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 the 1, 3, 11, 23, and 29 Articles, 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. Tho
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 trials is
6. This is not a just conclusion (Art. 19); for, it would follow, by the same mode of reasoning, that in six trials a person could not fan to throw an ace. The error, which is not easily seen, arises from lacit supposition that there must necessarily be a second trial, whicb is not the case if an ace be thrown in the first.
ON LIFE ANNUITIES.
(1.) To find the present value of an annuity of £i, to be continued during the life of an individual of a given age, allowing compound interest for the money.
Let , 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
B aumber left at the end of 1, 2, 3, &c. years' ; then is the value
A C D of the life for one year,
&c. its value for 2, 3, &c. years ;
Α' Α' 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 ; but it is
only to be paid on condition that the annuitant is alive at the end of
B the year, of which event the probability is therefore the pre
B sent value of the conditional annuity is
in the Ar
с ner, the present value of the second year's annuity is the pre
sent value of the third year's annuity is
&c. therefore the
whole value required is Àx++&c.)
to the end of
(2.) 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 above ten, as will appear from an inspection of the tables ; and on this supposition, the sum of the series 1/B с D
+ -+ +&c.)may be found. 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
n-2 0-3 one dies every year; and
&c. will be the prodabilities of his living 1, 2, 3, &c. years : hence, the present value of
7-1 n–2 , n-3 an annuity of £i to be paid during his life is + +
(x-1).x +&c. continued to n terms. The sum of the series
(n-1):-nx+x+1 to n terms, was found to be
n.(1-2) n-1 -2 -3
-+ + -+ &c. ton nr
, and the sum of the series
; the present value of the annuity.
(3.) Cor. 1. This expression for the sum is the same with
an annuity of £i to continue certain for n years, then P=
(4.) Cor. 2. The present value of the annuity to continue for ever, from the death of the proposed individual, is
n.(r-1) For, the whole present value of the annuity to continue for ever, is
and if from this its value for the life of the individual be
taken, the remainder
is the present value of the annuity to
n.(1-1) continue for ever, from the time of his death.
(5.) To find the present value of an annuity of £1, to be paid as long as two specified individuals are both living.
Find 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
b a, u, c, &c. and r the amount of £i in one year ; then + &c. is the present value of the annuity required.
(6.) To find the present value of an annuity of £1, to be paid as long as either of two specified individuals is living.
Find 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,
A B С then the present value of the annuity is +
++&c. (7.) Cor. If the annuity be M £, the present value is M times as
A B great as in the former case, or Mx (
+ &c. (187.) These are the mathematical principles on which the values of annuities 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, Mr. Baron Maseres, Mr, Morgan, in the Philosophical Transactions nd Dr. Waring.
METHOD OF DIFFERENCES.
Definitions. 1. A variable quantity is one that is continually changing its value, either by increasing or decreasing its magnitude.
2. A constant quancity is one that never changes its value.
3. The function of a variable quantity which changes by certain steps, is called an integral.
4. The quantity from which the function is raised is called the base of that function, or of that integral.
5. The difference between two similar functions of two quantities, of which the one to be taken away has any variable quantity, x for its base, and the other function from which it is taken, has the same base increased by a given quantity, is called the difference or increment.
Notation. 1. Invariable quantities are represented by the first letters, a, b, c, &c. of the Alphabet.
2. The base of the function of a variable quantity, is represented by r, and is supposed to be continually increased, at every step, by unity, which is, therefore, the difference or increment of x.
3. The differences of quantities which are supposed to be functions of I, are represented by the same letter, which indicates that function with I not under a.
Thus, suppose x to be a function of x, then z is the difference or increment of z.
4. The operation of taking the differences of two variable quantities is denoted by A, and if the same operation be performed on the difference, supposing it variable ; then the double operation is denoted by A; and if n successive operations are performed, then the syinbol denoting these n operations is A.
Axioms. 1. A constant quantity has no difference or increment.
2. The difference of any whole quantity, is the sum of the differences of all the terms, or parts of that quantity.
Problem 1. (1.) To resolve an algebraic product, consisting of binomial factors,