« PreviousContinue »
ON SOME QUESTIONS IN THE CALCULUS OF PROBABILITIES,
AND ON THE DETERMINATION OF MEAN VALUES.
SECTION 1.- On the Calculus of Probabilities. 264.] The Calculus of Probabilities contains two or three classes of problems, which cannot be solved without the Integral Calculus; and as they afford a good and apt illustration of definite integration, and are moreover in themselves curious and instructive, it is desirable to devote a few pages to their treatment. It is not of course my intention to enter on a discussion of the difficulties, metaphysical and moral, in which the calculus of probabilities is often enveloped. I shall assume and merely state the mathematical principles of it, which are few and simple, and apply them to certain problems which require definite integration.
The first principle, which is indeed the foundation of the calculus, is the following;
In a system of possible events, of which all are equally probable; that is, in which we know no reason why any one should occur rather than another; the probability in favour of any one is the ratio of the number of events favourable to its occurrence to that of all the events; and the probability against any one is the ratio of the number of events adverse to its occurrence to that of all the events.
The word Chance is synonymous with Probability; so that the preceding principle is valid when the word Probability is replaced by the word Chance.
Hence we have the following definitions ;
The chance or probability in favour of an event is the ratio of the number of favourable events to that of all the events.
The chance or probability against an event is the ratio of the number of unfavourable events to that of all the events.
And thus the mathematical definitions take the form of fractions.
The probability in favour of an event is a fraction of which
the numerator is the number of the favourable events, and the denominator is that of all the events.
The probability against an event is a fraction of which the numerator is the number of the unfavourable events, and the denominator is that of all the events.
The sum of these probabilities is unity; and as one or other of the possible events must (as it is assumed) occur, the result is certainty. Hence unity is the mathematical definition of certainty.
This result is also otherwise evident: in the case of certainty all the events are favourable, so that the numerator and the denominator of the fraction which defines the favourable chance are equal; and consequently the favourable chance, which is certainty, is unity.
The ratio of the number of favourable events to that of unfavourable events is called the odds in favour of an event.
The ratio of the number of unfavourable events to that of favourable events is called the odds against an event.
Hence it appears that probabilities and odds do not vary, when the number of possible events varies, if the ratio of the numbers of favourable and unfavourable events remains the same.
All these definitions may be expressed mathematically in the following manner; let A and B be two contradictory events; that is, B occurs if A does not occur; then, if a = the number of possible events favourable to the occurrence of A, and b = that of possible events favourable to the occurrence of B, all being equally probable,
The chance in favour of A =
265.] Now the case to which I propose first to apply these principles is that in which the number of possible events, as also that of the favourable events, is infinite; in which however the numbers of, or the areas or the extents which include these several classes of, events are not equal; so that these are to be calculated;
and they become the numerators or denominators of the fractions which express the chances and the odds. In these problems the several events are no longer discreet cases, as distinct units or integers; of which the sum is found by simple addition ; but they are elements of a continuous series, and their sums are definite integrals of which the limits are assigned by the conditions of the question. From the preceding principles we have the following definitions ; The chance in favour of an event
The extent of favourable cases
The extent of all possible cases' The chance against an event
The extent of unfavourable cases
The extent of all possible cases ! and similar definitions are of course true of the odds for and against an event. Although the direct solution of problems of this kind requires the integral calculus for the determination of these several extents, yet in many cases these can be compared without that process, and I propose first to give some examples of this kind.
Ex. 1. A needle is thrown on a plane horizontal table. What is the chance that the line of it is within 15° of the meridian?
In this example the whole extent of possible cases is 360°; and as the needle may lie within 15° of the meridian to the east or to the west, and either towards the north or the south, the extent of favourable cases = 60°. Hence The chance in favour of the event = 20 = a:
60 1 Ex. 2. A plane surface of indefinite area is divided into equal squares, and a thin circular coin falls flat on the surface. What is the probability that it does not fall across one of the dividing lines of the surface ?
Let a = the side of each square, and r = the radius of the coin. As all the squares are equal, it is evident that the estimation of the chance for any one square is sufficient for that of all. Draw lines parallel to the sides of a given square, and within it, at a distance = r from each side ; then a new square is described within the original one, each side of which = a-2r, and of which consequently the area =(a-2r)?. Now if the centre of the coin falls within this area, the coin does not intersect a dividing line of the area, and consequently the area of favourable cases = (a–2r)2. But as the centre of the coin may fall anywhere within the square, the area of all possible cases = ao. Hence
(a— 2r)2 The chance of the coin falling clear of a dividing line=
a> The chances may be similarly determined if the plane surface is divided into equilateral triangles or into regular hexagons.
Ex. 3. A circular coin falls flat on a circular plate; determine the chance that (1) the coin does not meet the edge of the plate; (2) the coin meets the edge of the plate and does not fall over; (3) the coin meets the edge and falls over.
Let R = the radius of the plate, r = the radius of the coin ; then it is evident that we may estimate only those areas on which the centre of the coin may fall. Now in each of the cases the area of possible events = 7 (R+r); and the areas of the favourable events are respectively 7 (R—r)?, ar (2R-r), ar (2 R+r); so that if y, cg, Cg are the chances,
2 Rr + 22 9 = 7732; C = *(R+rja cs = TR+rja As these three cases exhaust all the possible events, C, +Cg+Cz=1.
Ex. 4. A heavy spherical ball falls through a grating, the bars of which are infinitesimally thin, and are equidistant. What is the chance that it falls clear of a bar?
Let 2a = the distance of the bars from each other, and let r = the radius of the ball; then the chance = " .
Ex. 5. Four points are taken at random in a plane of indefinite area. What is the chance that some one of the four points is included within the triangle of which the other three points are the vertices ?
With these four points four triangles may be formed, let A, B, C be the three points which form the largest triangle, and let p be the fourth point. Through A, B, C respectively draw the lines B'C', &A', A'B' parallel to BC, CA, AB respectively; thus forming a new triangle a'B'c', the area of which is four times that of ABC, and so that x'BC = B'ca = c'AB = ABC. Now P must fall within the area a'B'd'; for if it fell without it, the triangle ABC would not be the greatest which could be made out of the original four points. Consequently A'B'c is the area of all possible events ; PRICE, VOL. II.
that is, of events consistent with the asumption that ABC is the largest triangle; and abc is the area of all the favourable events; and hence
the area of ABC The required chance =
the area of A'B'C'
The solution of this problem is due to Professor Cayley of Cambridge.
Similarly, if five points are taken at random in space, the chance that the fifth point may be included within the tetrahedron of which the other four points are the vertices =
266.] The following are examples in which the integral calculus is directly required.
Ex. 1. A large plane area is ruled with equidistant straight lines; a thin straight needle, the length of which is less than the distance between two consecutive lines, falls on the plane; what is the probability that it falls clear of a dividing line ?
In this problem we may evidently confine our attention to the space contained between two consecutive lines, and to the middle point of the needle with reference to that space.
Let 2a = the distance between two consecutive lines; and let 2c be the length of the needle. Let the distance between two consecutive lines be bisected by a straight line parallel to each of the lines, and let x = the perpendicular distance of the middle point of the needle in any position from this line; also let o be the angle at which the needle is inclined to the line perpendicular to the dividing lines. Now it is evident that x admits of all values from – a to a, and that o admits of all values from – a to Tt; and that the needle does not intersect a dividing line so long as x+c cos o is less than a, or —&—ccos o is greater than -a. Let dx be an infinitesimal-increment of x, which we will suppose to be coincident with the centre of the needle, and let debe an infinitesimal increment of $, and we will suppose do to be coincident with the needle. Then
The extent of all possible cases = [° /* dø dx