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Hence the chance that the needle is clear of a dividing line

2c

= 1; and the probability that it falls across a dividing

line =

πα

2 c

па

If in the calculation of the favourable extent, the p-integration

had preceded the x-integration, then, if • = cos−1

The extent of the favourable cases

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C

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which is the same result as before.

The result of this problem suggests a curious way of finding the value of π. Let a needle of given length be thrown on a table ruled as in the question; and in m+n trials let m = the number of times in which the needle falls clear of a dividing (m+n)2c

line; then =

2 c
n
па m + n

; so that π =

na

Ex. 2. In the adjacent sides OA, OB of a rectangle, two points P and Q are taken at random; what is the probability that the rectangle contained by op and oq is not greater than the nth part of that contained by OA and OB?

Let OA = a, OB = b; OP = x, oq = y; then

The extent of all possible cases =

α b

SLdy dr

0 0

= ab.

In calculating the favourable cases, the limits of a and y are given by the inequality -xy>0; so that for all values of x

a

ab

n

between and 0, y ranges from 6 to 0;

n

but for any value of ab

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to 0; consequently

пх

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The following is another solution of this problem;

Let OA and Oв be taken as rectangular axes of x and y respectively, and let the rectangular hyperbola, whose equation is ab

xy== be drawn; and let it intersect the sides of the rect

n

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angle opposite to OA and OB in C and D respectively; let R be the point (x, y); and from R draw OP and oQ perpendicular to oA and OB respectively; then the conditions of the problem are fulfilled if R falls within the area OADCB.

dx

Now the probability that P falls between x and x + dx = ;

a

and that Q falls between y and y + dy = dy; consequently the

b

dy dx probability of the concurrence of these two events = ; and ab this is the probability that R falls within the area dy dr. Hence,

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Ex. 3. A stick is broken into three parts; what is the chance that the sum of the lengths of every two is greater than the length of the third; that is, that the three parts will form a triangle?

=

Let the length of the rod 2a; and let x = the length of the first part, and y = the length of the second part. Then

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In calculating the favourable cases a may range from a to 0;

and since x+y>a, y may range from a to a-x; so that

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Ex. 4. Two arrows are fixed in a circular target. What is the probability that the distance between them is greater than the radius of the target?

Let the radius of the target = a; and let 2r = the distance of one of the arrows from the centre of the target; so that 4r dr de is the area-element of the target which we may suppose to be occupied by this arrow. Now the area of that part of the target, all of whose points are at a distance from this arrow not less than the radius of the target, = 2r (a2 — r2)3 +2a2sin Consequently the extent of the favourable cases

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r

a

Ex. 5. Two arrows are fixed in a circular target. What is the probability that they and the centre of the target are at the angles of an acute-angled triangle?

and let P, the place of one Let o be the centre of the

Let a the radius of the target; of the arrows, be at the point (r, 0). target; join OP, and on OP as a diameter describe a circle; a* o and P draw lines perpendicular to op; then the area of the part of the target included between these two lines

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and if q, the place of the other arrow, is in any point of this

area, except within the small circle, the triangle orq is acuteangled. Hence

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The extent of all the cases =
= [" ["=a2r dr de

Ο

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As this result is independent of the radius of the target, it is true when that radius is infinite, in which case the centre is an indeterminate point; and hereby we have the conclusion that in infinite circular space, if three points are taken at random, one of them being the centre of the space, the probability that they are the angles of an acute-angled triangle is

1

4

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267.] The integral calculus is also required for the complete discussion of the laws of combination of events, when the number of events is infinite; and this inquiry is most important in the indirect application of the theory of probabilities to the investigation of the probabilities of the causes to which certain observed events are due; because in most physical investigations, the number of events or of, what amounts to the same thing, observations may be infinite, and it is necessary to ascertain the character which this circumstance brings into the result. Here however I must first have recourse to some elementary considerations of compound events.

Let there be a system of possible events of only two classes; one of which produces A, and the other produces B; and let a and b be the numbers of events of these two classes respectively; and let us suppose all to be equally probable, and assume the b necessity that either A or B must occur. Thus

a

a+b'a+b

the probabilities in favour of A and B respectively. Let

Α

are

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so that p and q are the probabilities in favour of A and B respectively. Also, it is evident that

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so that q = 1-p; and the sum of the probabilities

represents certainty.

(8)

= 1, which

Now let us suppose m trials, or observations, to be made on this system of possible events; then the probability that

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these are all the possible combinations of the events, and are evidently the successive terms of the development of a binomial; consequently the sum of all the probabilities = (p+q)m = 1.

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Hence it appears that of the several terms of (p+q) in its expanded form each expresses the probability of an event compounded of a repeated as many times as is the index of p, and of B repeated as many times as is the index of q. Also that the sum of the first n+1 terms gives the probability that in m trials A will occur not less than m-n times; or, what is the same thing, that B will occur not more than n times.

m

Since the number of terms of (p+q)" is m+1, the number of terms increases as m increases; but as (p+q) = 1, the sum of the terms is always the same, and consequently each term decreases as m increases. Thus the probability of a particular combination becomes less and less, although the sum of all the probabilities is the same.

These results admit of the following graphic representation. Take a straight line oa of definite length, in the axis of x, say; and divide it into m equal parts; at each of the m+1 points of partition draw ordinates in order, severally proportional to the several successive terms of the development of (p+q)TM, and join their extremities; we shall hereby have a broken line, which will ultimately become a curve under certain conditions, when m = ∞, the ordinates of which represent the probabilities of certain compound events which are assigned by the abscissæ.

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