786 PROBABILITY Hence, for the circle, the probability is a maximum or minimum. It will be a minimum, because in the formula (68) for the mean triangle formed by three points (M -M)A = 3a(M 1 -M). ilj, which is the meau triangle when one point is in a, is really greater than when it is on the circumference, though the same in the limit ; hence Mi>iM; .-. (M -M)A>oM; .-. M /(A + a)>M/A. Therefore, if we consider infinitesimals of the second order, the chance of a re-entrant figure is increased by the addition of the space a to the circle. It will be an exercise for the reader to verify this when the space is subtracted. For an ellipse, being derived by projection from the circle, the probability is the same, and a minimum. It is pretty certain that a triangle will be found to be the con tour which gives the probability the greatest. Mr Woolhouse has given (Educ. Times, Dec. 1867) the values of p for 73. Many questions may be made to depend upon the four-point problem. Thus, if two points A, B are taken at random in a given convex area, to find the chance that two others C, D, also taken at ran Join, shall lie on opposite sides of the line AB. Let p be the chance that ABCD is re-entrant. If it is, the chance is easily seen to be that any two of the four lie on opposite sides of the line joining the two others. If ABCD is convex, the same chance is ^ ; hence the required probability is Triangle. Parallelogram. Reg. Hexagon. Circle. J or -3333 u 3056 289 -5TT 2973 M*- 2 2955
Or we might proceed as follows, e.g. , in the case of a triangle : The sides of the triangle ABC (fig. 5) produced divide the whole triaugle into seven spaces. Of these, the mean value of those marked a is the same, viz., the mean value of ABC, or T J 5 of the whole triangle, as we have shown, the mean value of those marked being f of the triangle. This is easily seen : for instance, if the whole area = i, the mean value of the space PBQ gives the chance that if the fourth point D be taken at random B, shall fall within the triangle ADC ; now the mean value of ABC gives the chance that D Fig. 5. shall fall within ABC ; but these two chances are equal. Hence we see that if A, B, C be taken at random, the mean value of that portion of the whole triangle which lies on the same side of AB as (. does is r j of the whole, and that of the opposite portion is T V Hence the chance of C and D falling on opposite sides of AB is T V 74. We can give but few of the innumerable questions depend ing on the position of points in a plane, or in space. Some may be solved without any aid from the integral calculus, by using a few very evident subsidiary principles. As an instance, we will state the following two propositions, and proceed to apply them to one or two questions : (1) In a triangle ABC, the frequency of any direction for the line CX is the same when X is a point taken at random on the base AB as when X is taken at random in the area of the triangle. (2) If X (fig. 6) is a point taken at random in the triangle AB6 (Bb being infinitesimal), the x . frequency of the distance , . , I AX is the same as that of A Y z B AZ, Y and Z being two Fi g- 6 - points taken at random in AB, and Z denoting always that one nf the two v:hich is nearest to B. For the frequency in each case 13 proportional to the distance AX or AZ. Let us apply these to the follow ing question : A point is taken at random in a triangle (fig. 7) ; if n more points arc taken at random, to find the chance that they shall all lie on some one of the three triangles AOB, AOC, BOC. If C be joined with all the points in question, every joining line is equally likely to be nearest to CB. Fig. 7. Hence the chance that all the n points fall on the triangle ACD is If this is so, we have to find the chance that all lie on AOC. Now if range over the infinitesimal triangle DCd, we may, by prin ciple (2) above, suppose it to be the nearest to D of two points taken at random in CD. If so, the chance that AO is nearer to AD than any of the lines from A to the n points is 2(?H-2)- 2 ; for, by (1) above, we may suppose all the points taken at random in CD; now any one of the + 2 is equally likely to be the last ; and is the last of the two additional points. Hence, if is in the triangle CDd, the chance that the n points fall on AOC is therefore this is the chance wherever falls in ABC. Therefore the required chance that the n points fall on some one of the triangles AOB, AOC, BOC is Again, if be taken at random in the triangle, and three more points X, Y, Z be also taken at random in it, to find the chance that they shall fall, one on each of the triangles AOB, AOC, BOC. First, two of the points are to fall on one of the triangles ACD, BCD, and the remaining one on the other ; say two on ACD, the chance of this is j, as CO must then be the third in order of the four distances from C. If this is so, the chance that the point X in BCD falls on BOC is . For, as above, if ranges over the triangle CDd, we may take it to be the lowest of two points taken at random on CD ; and the chance that, if another point be also taken at random in CD, it shall be lower than is $. Now if one of the points X is in BOC, the frequency of in CDd will be the same as that of the lowest of three points taken on CD; and the chance that one of the remaining points shall fall in AOC and the other in AOD is the chance that 0, the lowest of three particular points out of five, all tnken at random in CD, shall be the fourth in order from C. It is easy to see that this chance is T V Hence the chance that one point falls on BOC, one on AOC, and the third on AOD is 12 S _ 1 4 5 TT TV And it will be the same for the case where the third falls on BOD. Hence the chance that one point falls on each of the three triangles above is double this, or ^V 75. Straight lines falling at random on a Plane. 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 coordinates p, u, the perpen dicular on it from a fixed origin 0, and the inclination of that perpendicular to a fixed axis, then, if p, u 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 a> between and ca + du, will be measured by dpdta, and the integral between any limits, measures the number of lines within those limits. It is easy to show from this that the number of random lines which meet any closed convex contour of length L is measured by L. For, taking inside the contour, and integrating first for p, from to p, tho perpendicular on the tangent to the contour, we have fpd<a ; taking this through four right angles for u, we have by Legendre s theorem on rectification, N being the measure of the number of lines, N= /" 2 V?o. = L. 1 yo Thus, if a random line meet a given contour, of length L, the chance of its meeting another convex contour, of length I, internal to the former, is p-J/L. If the given contour be not convex, or not closed, N will evi- i 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 I 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 $L, half the length of the boum aiy. If we fake 2/ as the measure for the liiiCj the measure for the contour will be L, as above. Of course we have to remember that each line must meet the contour twice. 1 would bo possible to rectify any closed curve by means of this principle. Suppose it traced on the surface of a circular disk, of circumference L, 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 ot
the circlfc.