1 ces; that is, the probability of the compound occurrence is equal to the products of the partial probabilities. For as each of the m, may concur with each of the m, cases, there will be m1×m, possible cases, which, by the supervening of m, new cases, increase to m1×m2 × m3, and so on. The same reasoning applies to the favorable cases g1, ga, g3, &c., from whence, by the principles already established, results formula (III.). Let it be required, for example, to draw out of a vase which contains the numbers 1, 2, 3, 4, 5, and 6, first 1, then either 2 or 3, and, finally, 4, 5, or 6, in three drawings; the probability is expressed by If the partial occurrences are equal (that is, repetitions of the same), then =()". Thus, if with each of three dice, 6 shall be thrown, w= IV. Should there be m possible cases, of which g are favorable and u unfavorable, and of these k+r are to occur, so that k of the favorable, with r of the unfavorable, must come in juxtaposition, then the expression for the probability of the occurrence of every such order is This depends on (III.), each of the factors in the above value of w expressing the partial probability of the single occurrence of a 1st, 2d, ....kth favorable case, also of a 1st, 2d, ....rth unfavorable case, and the product expressing the probability of these occurring in a certain order. EXAMPLE. If from 20 tickets, 8 of which are prizes and 12 blanks, 6 are to be drawn ; then, in favor of the requisition that exactly two prizes shall be first drawn, or shall occupy any given place in the order, w= 77 (W×HHH(=3230" V. Should there be required in the supposition of the last case no particular order for the single cases which occur, the expression becomes g-k-1 m (( u-r+1 1). ... (V.) Thus it will be found that, if from 30 appointed numbers out of 90, 5 of the whole 90 are to be drawn, so that just 3 of the 30 shall be among those drawn, it being immaterial at which three of the five drawings, the expression for the probability in this case is 20= VI. Should the number of possible cases continue to remain the same, while the other circumstances are as in (V.), the formula would be EXAMPLE. The probability of throwing the same face three times in 7 casts of a die, or one cast of 7 dice, would be expressed by VII. Let the probability be required that of two different occurrences the first, or, if this does not, the second, shall happen; if the single probability of the first happening be expressed by w, the probability of its failing will be expressed by 1-w; this must be combined with the probability of the second happening, according to (III.), giving (1-w1)w2 for the probability of the second happening, if the first fails: then the compound probability required is expressed (II.) by w=w1+w2(1—w1)=w1+w2-W1 • W2• EXAMPLE. Required the probability of throwing with two dice, at the first cast 8, and, if this does not happen, 9 at the second cast. 5 +36 (136) = 36 + 36 36 -81 36 VIII. Above we have considered the absolute probability of the happening of an event; the relative probability of the happening of two events is expressed by the formula The relative probability of throwing with two dice rather 7 than 10, is ex6 2 w1+w26+3-3 pressed by IX. When money depends on the happening of an event, the product of the sum risked, multiplied by the expression for the probability of the event on which it depends, is called the mathematical expectation. If there be among m+m cases, my favorable for one party, and m, for the other, the sum risked by the first a,, and by the second ag, then for the mathematical expectation of each we have (1) M2 m1 + m2 Therefore, when ee, it is necessary that a,: a=w1: w2. This principle is important in the subject of annuities and life insurance. For its application, and that of all the foregoing theory to which, see De Morgan on Probabilities. EXAMPLES. (1) How many binary combinations of oxygen, hydrogen, nitrogen, carbon, sulphur, and phosphorus? How many ternary combinations of the same? (2) How many combinations of 5 colors among those of the prism, viz., red, orange, yellow, green, blue, indigo, and violet? * 12 and 2 can each be thrown with two dice but in one way, 11 and 3 each in two ways, 10 and 4 in three ways, 5 and 9 in four ways, 6 and 8 in five ways, 7 in six ways. (3) What is the probability of throwing with three dice two equal numwith five dice, three equal? bers (4) What of throwing with two dice the faces 2, 4, and 6? (5) What the probability that a dollar tossed twice will fall head up once? (6) Of which is the probability greater, the drawing at three trials from 52 cards three cards of different colors, of which there are four, or three face cards, of which there are 12? (7) What of drawing out of a vase containing 5 white, 6 red, and 7 black balls, in two drawings, 2 red, or else a white and a black ball? (8) What of drawing out of the same vase, in three drawings, 3 of different colors, or else 2 black and 1 white? (9) What of throwing with four dice 15, or with three dice 12? METHOD OF UNDETERMINED COEFFICIENTS. 209. The method of undetermined coefficients is a method for the expansion or development of algebraic functions into infinite series, arranged according to the ascending powers of one of the quantities considered as a variable.* The principle employed in this method may be stated in the following THEOREM. If Axa+Bxß+Cx¥ +, &c., =A'xa'+B'xß'+C'x¥'+, &c. (1), for all values of r, then must the exponents of x in the two members be the same, and the coefficients of the same powers of x the same. For, dividing (1) by ra, we have A+BxBa+CxY—a+, &c., =A'xa¬a+B'xß'¬a+C'x?'¬a+, &c. (2) Since may have any value, make it zero; the first member thus reduces to A, while the second becomes zero, unless we suppose a equal to some one of the exponents a', B', y', .... Suppose it to be a'. Then we have a=a', and .. A=A'. Suppressing the equal terms A and A'ra'-a from the two members of (2), and dividing it by x-a, it becomes B+Cx-3+, &c., =B'x-ß+Cx¬3+, &c. Making, again, x=0, the first member reduces to B, and the second to zero, which is absurd, unless we make ẞ equal to some one of the exponents of x, say ẞ', in the second member, and then B=B'. Proceeding in this way, the exponents of x, and the coefficients of the same powers of r in the one member, may be proved equal to those in the other. The above theorem may be expressed in a modified form; thus, if all the terms of (1) be transposed to the first member, it becomes, collecting the equal powers of x, a and a', ẞ and ẞ', &c., (A-A')x+(B-B')x+(C-C')x+, &c., =0; from which, since A=A', B=B', &c., we perceive that when a function of r is equal to zero for all values of x, the coefficients of the different powers of r are equal to zero separately. A variable quantity is one which is either entirely indeterminate, so that it may have any value at pleasure, or one which varies in conformity with certain conditions imposed. in which some of the coefficients A, B, C, &c., may be zero, and thus certain 1=A+ Br+ Cx2+ Dr3+ Ex1+.... Hence, by the preceding theorem, we have 1 Therefore 1-2x+x ;=1+2x+3x2+4x2+5x+6x+...... The equality of a function to a series is hypothetical; and after A, B, C,...... have been found, the result must be carefully examined. If we put the func tion 1 1 3x -x2 3x-x2 - =A+Bx+, &c., it gives the absurdity -1=0. We must put =Ax¬1+Bxo+Cx+Dx2+, &c. The method of indeterminate coeffi cients is to be avoided where other methods will apply. Assume √1+x=A+Bx +Cx2 +Dr3+..., and square both sides; Hence, equating the coefficients of the like powers of x, we have A into two fractions having simple binomial de By quadratics we find x2-13x+40=(x−5)(x−8); hence we may assume 3x-5 = .. 3x-5 (A+B)x-(8A+5B); and by the principle of undetermined coefficients we have A+B=3, and 8A+5B=5. Note. The values of A and B might have been determined in the following Now this equation must subsist for every value of r; and therefore, This method may frequently be employed with advantage, and will be found useful in the integration of rational fractions, in the Integral Calculus. When the denominator is composed of equal factors, such as (x+a)3, (x—b)2, it will be necessary to assume the given function equal to |