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tion, the probability, infinitesimal indeed, of the action of the cause which produces the event A, and of which the probability 2010 (1-x)" dx

(22) 1 m (1 — X)" dx

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272.] Hence we can easily deduce the probability of the future event A. As (22) gives the probability of the hypothesis which assigns to a the particular probability x, the probability of the occurrence of that event

xm+1 (1 — x)" dx

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and the sum of all these probabilities, as x varies from 0 to 1, will give the whole probability of the event A. Hence

'æm+1 (1 — x)" dx the probability of a =

(25)
| 2m (1-x)" dx
B(M+2, n+1)
B(m+1, n+1)
r(m+2) r(n+1) r(m+n+2)

r(m+n+3) T(m+1)r(n+1)
m+1

(26) m+n+2

n+1 Similarly the probability of B =

(27)

m+n+2 The sum of these probabilities = 1, as the result ought to be, since the events are contradictory.

If the event a has occurred m times consecutively, and B has not occurred at all, n = 0; so that

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Thus, for example, if the Sun has already risen m times, and it has not been observed not to rise, the probability of the Sun's rising again =

in m +1.

m + 2 which = 1, that is, certainty, when m = 0.

273.] When of two contradictory events A and B, A, of which the probability = x, has already occurred m times, and B, n times, the probability that in the next (p+q) times a will occur p times and B q times may thus be found.

By reason of Art. 267 the probability of the required compound event (p+q) (P+91) ... (p+1) 2P (1-x)";

1.2.3. ...a and multiplying this by the probability of the hypothesis which is given in (22); and taking the sum of all the products thus formed for the limits 0 and 1 of x, the required probability

A zem+P (1 — 2)"+9 de
_ (P+9) (P+q-1) ... (p+1) JO
1.2.3. ...a

20" (1 — 2)" dx
r(p+q+1) r(m+p+1) r(n+q+1) r(m+n+2)

· (29) r(p+1) r(q+1) r(m +1) r(n+1) r(m+n+p+9+2) If n = q = 0, then the probability that a will occur p times in succession, when it has already occurred m times, r(m+p+1) r(m+2)_ m+1

(30) T(m+ 1)r(m+P+2) = m+p+1' Thus, if a coin has turned up heads four times in succession, and no tail has occurred, the probability that it will turn up heads the following three times in succession =

If p = m, and m is very large, the probability that a will occur m times without interruption, when it has already occurred in times without interruption, = 5.

We may also hereby determine the value of x which gives the most probable of all the hypothetical causes which can produce A. Thus let p be the general probability; then by (22),

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(32)

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20m (1 -- x)" dx

(31) 1 209 (1 — 2)" dx Now the denominator of this fraction being a definite integral does not vary with x. Consequently if = 0, we have,

2017-1 (1-x)"-1 {m(1 — X) — nx} = 0;
m

n
.. X =-
min'

mtn Thus, the most probable of all the hypotheses is that in which the probabilities of the events A and B are equal to the ratios which the number of the favourable past events bear to the whole number of past events.

274.] From the preceding equations we can also deduce the following problem, which is of considerable interest. If in a certain number of observations in which two contradictory events A and B are possible, one, say a, has occurred more frequently than the other, we are naturally led to suspect the existence of some cause of this result. If the producing causes of the two were equal, one should in the long run occur as frequently as the other; but if this equality is interrupted, we suspect a preponderance of cause in favour of that event which more frequently occurs; that is, we suspect that the probability in favour of that event is greater

and the suspected preponderance increases according as the number of events of that particular kind increases. In this case if a is that event on the side of which the preponderance exists, then the limits of x, which is its probability, are 1 and so that if P is the probability of the existence of a cause which produces a, by (31)

ond

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If n = 0, so that A has occurred m times without interruption, and B has not occurred at all,

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which expresses the probability of the existence of a cause, that is, of an hypothesis the probability of which is greater than , which favours the repetition of A.

Thus, for example, all the eighty-eight planets which constitute the Solar system as at present (Oct. 1, 1864) known, have a direct motion ; that is, move round the Sun from west to east. And, as the motion of all these in the same direction is the repetition of the same fact, whereas the contrary fact might exist, it shews the high probability of the existence of a cause which produces this fact: and if p is this probability,

Ps 289

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,

which fraction nearly =1; and consequently the doctrine of chances shews the well-nigh absolute certainty of a physical cause of this fact.

The limits of my work preclude me from giving other problems in this most interesting branch of the higher mathematics, and I can only refer the reader to treatises where the subject is specially investigated.

SECTION 2.-On the Determination of Mean Values. 275.) When n different values are assigned to the variable of a function, so that the function thereby receives n values, the nth part of the sum of these values is called the mean or average value of the function, or in more precise terms, the arithmetical mean value of the function. This is the definition of mean value when the values of the function arise from discontinuous values of the variable ; but an analogous definition is also applicable when the variable varies continuously. In this case let us suppose f (x) to be a function of x which varies continuously, and does not become infinite, between the limits X, and Xo; and let us suppose X, —%, to be divided into n equal parts each of which = i; so that Xn—X, = ni; then the mean value of the functions corresponding to the several points of partition f (x) + f (xo+i) + f(x8+2i) + ... +f{«o+(n-1);}

· (35)

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Let the numerator and denominator be multiplied by i; and let us suppose x to increase continuously from X, to xm; then

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To

and this is the definition of mean or average value in its application to a continuous function. The numerator and denominator will be definite multiple integrals, if their elements are multiple differentials. The following examples are in illustration of their definitions,

Ex. 1. Find the mean distance of all points within a circle (1) from the centre; (2) from a point in the circumference, the radii vectores to the points being regularly distributed in each case.

The meaning of the last condition is, that the finite angle, through which the radii vectores drawn to the several points of the circular area extend, is divided into equal elements. Let a

= the radius of the circle; and in both cases let the point p from which the line is drawn be coincident with and expressed by the area-element; so that r dr do denotes that point. Then as r is the distance of this point from the origin, p2 dr do is the element of the definite integral which is the numerator, and r dr do is the element of the definite integral which is the denominator of the fraction whereby the mean value is determined.

27a3 In case (1), the numerator = 1 / 2 dr do = 0 ;

do do

27 pa
the denominator =1 / rdr do = a?;

Jo 10
is the required mean value = ? a.
In case (2), if r = 2 a cos 0,
The numerator = { ["r2 dr de
= **cos 0)3 do = "90-;

32a3.

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the denominator = / 1 r dr do = 7a;

32a in the required mean value = PRICE, VOL. II.

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3 D

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