Page:EB1911 - Volume 22.djvu/401

Now if the rod crosses one of the lines we must have c > x/cos θ; so that the favourable cases will be measured by

$\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}\,d\theta \int _{0}^{c\cos {\theta }}\,dx=2c$.

Thus the probability required is p = 2c/πa.

It may be asked—why should we take the centre of the rod as the point where distance from the nearest line has all its values equally probable? Why not one extremity of the line, or some other point suited to the circumstances of projection? Fortunately it makes no difference in the result to what point in the rod we assign this pre-eminence.

77. The legitimacy of the assumption obtains some verification from the success of a test suggested by Laplace. If a rod is actually thrown, as supposed in the problem, a great number of times, and the frequency with which it falls on one of the parallels is observed, that proportionate number thus found, say p, furnishes a value for the constant π. For π ought to equal 2c/pa. The experiment has been made by Professor Wolf of Frankfort. Having thrown a needle of length 36 mm. on a plane ruled with parallel lines at a distance from each other of 45 mm. 5000 times, he observed that the needle crossed a parallel 2532 times. Whence the value of π is deduced 3.1596, with a probable error^[1] ± .05.

78. More hesitation may be felt when we have to define a random chord of a circle,^[2] for instance, with reference to the question, what is the probability that a chord taken at random will be greater than the side of an equilateral triangle? For some purposes it would no doubt be proper to assume that the chord is constructed by taking any point on the circumference and joining it to another point on the circumference, the points from which one is taken at random being distributed at equal intervals around the circumference. On this understanding the probability in question would be ½. But in other connexions, for instance, if the chord is obtained by the intersection with the circle of a rod thrown in random fashion, it seems preferable to consider the chord as a case of a straight line falling at random on a plane. Morgan Crofton^[3] himself gives the following definition of such a line: If an infinite number of straight lines be drawn at random in a plane, there will be as many parallel to any given direction as to any other, all directions being equally probable; also those having any given direction will be disposed with equal frequency all over the plane. Hence, if a line be determined by the co-ordinates p, ω, the perpendicular on it from a fixed origin O, and the inclination of that perpendicular to a fixed axis, then, if p, ω be made to vary by equal infinitesimal increments, the series of lines so given will represent the entire series of random straight lines. Thus the number of lines for which p falls between p and p + dp, and ω between ω and ω + dω, will be measured by dpdω, and the integral ∬dpdω, between any limits, measures the number of lines within those limits.

79. Authoritative and useful as this definition is, it is not entirely free from difficulty. It amounts to this, that if we write the equation of the random line

x cos α + y sin α − p = 0,

we ought to take α and p as those variables, of which, the equicrescent values are equally probable—the equiprobable variables, as we may say. But might we not also write the equation in either of the following forms

(1)	x/a + y/b − 1 = 0,
(2)	ax + by − 1 = 0,

and take a and b in either system as the equiprobable variables? To be sure, if the equal distribution of probabilities is extended to infinity we shall be landed in the absurdity that of the random lines passing through any point on the axis of y a proportion differing infinitesimally from unity—100%—are either (1) parallel or (2) perpendicular to the axis of x. But the admission of infinite values will render any scheme for the equal distribution of probabilities absurd. If Professor Crofton's constant p, for example, becomes infinite, the origin being thus placed at an infinite distance, all the random chords intersecting a finite circle would be parallel!

80. However this may be, Professor Crofton's conception has the distinction of leading to a series of interesting propositions, of which specimens are here subjoined.^[4] The number of random lines which meet any closed convex contour of length L is measured by L. For, taking O inside the contour, and integrating first for p, from 0 to p, the perpendicular on the tangent to the contour, we have ∫pdω; taking this through four right angles for ω, we have by Legendre's theorem on rectification, N being the measure of the number of lines,

N − $\int _{0}^{2\pi }p\,d\omega$ = L.^[5]

Thus, if a random line meet a given contour, of length L, the chance of its meeting another convex contour, of length l, internal to the former is p = l/L. If the given contour be not convex, or not closed, N will evidently be the length of an endless string, drawn tight around the contour.

Fig. 1.

81. If a random line meet a closed convex contour of length L, the chance of it meeting another such contour, external to the former, is p = (X − Y)/L, where X is the length of an endless band enveloping both contours, and crossing between them, and Y that of a band also enveloping both, but not crossing. This may be shown by means of Legendre's integral above; or as follows:—

Call, for shortness, N(A) the number of lines meeting an area A; N(A, A′) the number which meet both A and A′; then (fig. 1)

N(SROQPH) + N(S′Q′OR′P′H′) = N(SROQPH + S′Q′OR′P′H′) + N(SROQPH, S′Q′OR′P′H′),

since in the first member each line meeting both areas is counted twice. But the number of lines meeting the non-convex figure consisting of OQPHSR and OQ′S′H′P′R′ is equal to the band Y, and the number meeting both these areas is identical with that of those meeting the given areas Ω, Ω′; hence X = Y + N(Ω, Ω′). Thus the number meeting both the given areas is measured by X − Y. Hence the theorem follows.

82. Two random chords cross a given convex boundary, of length L, and area Ω; to find the chance that their intersection falls inside the boundary.

Consider the first chord in any position; let C be its length; considering it as a closed area, the chance of the second chord meeting it is 2C/L; and the whole chance of its coordinates falling in dp, dω and of the second chord meeting it in that position is

2C dpdω/∬dpdω = 2/L²Cdpdω.

But the whole chance is the sum of these chances for all its positions;

∴ prob. = 2L⁻²∬Cdpdω.

Now, for a given value of ω, the value of ∫Cdp is evidently the area Ω; then, taking ω from π to 0, we have

required probability = 2πΩL⁻².

The mean value of a chord drawn at random across the boundary is

M = ∬Cdpdω/∬dpdω = πΩ/L.

83. A straight band of breadth c being traced on a floor, and a circle of radius r thrown on it at random; to find the mean area of the band which is covered by the circle. (The cases are omitted where the circle falls outside the band.)^[6]

If S be the space covered, the chance of a random point on the circle falling on the band is p = M(S)/πr ², this is the same as

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1. As recorded by Czuber, Geometrische Wahrscheinlichkeiten, p. 90.
2. Cf. Bertrand, Calcul des probabilités, pp. 4 seq. The matter has been much discussed in the Educational Times. See Mathematical Questions . . . from the Educational Times [a reprint], xxix. 17-20, containing references to earlier discussions, e.g. x. 33 (by Woolhouse).
3. Loc. cit. § 75.
4. The whole of p. 787 of Morgan Crofton's article is often referred to, and parts of pp. 786, 788 are transferred here.
5. This result also follows by considering that, if an infinite plane be covered by an infinity of lines drawn at random, it is evident that the number of these which meet a given finite straight line is proportional to its length, and is the same whatever be its position. Hence, if we take l the length of the line as the measure of this number, the number of random lines which cut any element ds of the contour is measured by ds, and the number which meet the contour is therefore measured by ½L, half the length of the boundary. If we take 2l as the measure for the line, the measure for the contour will be L, as above. Of course we have to remember that each line must meet the contour twice. It would be possible to rectify any closed curve by means of this principle. Suppose it traced on the surface of a circular disk, of circumference, and the disk thrown a great number of times on a system of parallel lines, whose distance asunder equals the diameter, if we count the number of cases in which the closed curve meets one of the parallels, the ratio of this number to the whole number of trials will be ultimately the ratio of the circumference of the curve to that of the circle. [Morgan Crofton's note.]
6. Or the floor may be supposed painted with parallel bands, at a distance asunder equal to the diameter; so that the circle must fall on one.