PROBABILITY 787 dently be the length of an endless string, drawn tight around the contour. 76. If a random line meet a closed convex contour, of length L, the chance of it meeting __ _R another such contour, external to the former, is where X is the length of an endless band envelop ing both contours, and crossing between them, and Y that of a band also enveloping both, but P ~ P not crossing. This may Fig. 8. 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- 8) 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 ft, ft ; hence Thus the number meeting both the given areas is measured by X - Y. Hence the theorem follows. 77. Two random chords cross a given convex boundary, of length L, and area ft ; 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, d<a and of the second chord meeting it in that position is 2C ^dpdco __2^ L ffdpdo) L" But the whole chance is the sum of these chances for all its positions ; . . prob. = 2L - tffCdpdu . Now, for a given value of o>, the value of jCdp is evidently the area ft ; then, taking <a from ir to 0, required probability = 27r,flL~ 2 . The mean value of a chord drawn at random across the boundary is J/Cdpdca Trft M = ^ --JT- . JJdpda 78. 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. ) l If S be the space covered, the chance of a random point on the circle falling on the band is This is the same as if the circle were fixed, and the band thrown on it at H random. Now let A (fig. 9) be a position of the random point ; the favourable cases are when HK, the bisector of the band, meets a circle, -p- Q centre A, radius |c ; and the whole number are when HK meets a circle, centre 0, radius r + Jc; hence the probability is 2r + c This is constant for all positions of A ; hence, equating these two values of p, the mean value required is The mean value of the portion of the circumference which falls on the band is the same fraction of the whole circumference. 1 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. If any convex area whose surface is ft and circumference L be thrown on the band, instead of a circle, the mean area covered is For as before, fixing the random point at A, the chance of a random point in ft falling on the band is jj = 27r. Jjc/L , where L is the perimeter of a parallel curve to L, at a normal distance ^c from it. Now M(S)_ jrc_ 79. Buffon s problem may be easily deduced in a similar manner. Thus, if 2r = length of line, a distance between the parallels, and we conceive a circle (fig. 10) of diameter a with its centre at the middle of the line, 2 rigidly attached to the latter, and thrown with it on the parallels, this circle must meet one of the parallels ; if it be thrown an in finite number of times, we shall thus have an infinite number of chords crossing it at random. Their number is measured by 2?r . a, and the number which meet 2r is measured by 4r. Hence the chance that the line 2r meets one of the parallels is 2? = 4r/Trrt . 80. To investigate the probability that the inclination of the line joining any two points in a given convex area ft shall lie within given limits. We give here a method of reducing this question to calculation, for the sake of an integral to which it leads, and which is not easy to deduce otherwise. First let one of the points A (fig. 11) be fixed ; draw through it a chord PQ = C, at an inclination to some fixed line; put AP = r, AQ = r ; then the number of cases for which the direction of the line joining A and B lies between and 6 + dd is measured by Now let A range over the space be tween PQ and a parallel chord distant dp from it, the number of cases for which A lies in this space and the direction of AB is from 9 to 6 + de is (first considering A to lie in the element drdp) Idpdef (r 2 + r 2 )dr = Let p be the perpendicular on C from a given origin 0, and let o> be the inclination of p (we may put dta for d6), C will be a given function of p, o> ; and, integrating first for o> constant, the whole number of cases for which u falls between given limits co , o> is J/~ W dw/C s dp ; J<a" J the integral f(Pdp being taken for all positions of C between two tangents to the boundary parallel to PQ. The question is thus reduced to the evaluation of this double integral, which, of course, is generally difficult enough ; we may, however, deduce from it a remarkable result ; for, if the integral be extended to all possible positions of C, it gives the whole number of pairs of positions of the points A, B which lie inside the area ; but this number is ft 2 ; hence the integration extending to all possible positions of the chord C, its length being a given function of its co ordinates p, (a. COR. Hence if L, ft be the perimeter and area of any closed c ivex contour, the mean value of the cub of a chord drawn across it at random is i>ft~/L . 81. Let there be any two convex boundaries (fig. 12) so related that a tan gent at any point V to the inner cuts off a constant segment S from the outer (e.g., two concentric similar ellipses) ; let the annular area between them be called A ; from a point X taken at random on this annulus draw tangents 2 The line might be anywhere within the circle without altering the question. 3 This integral was given by the present writer in the Comptes Rendus, 18<i9, p. 1469. An analytical proof was given by Serret, Annales scient. de CEcole Normale, 1869 p. 177.
12.