December 13, 1900] 



NATURE 



155 



E ; and so, of course, there must be a total number y of such 

 cases such that qy is an integer. 



In the problem before us let Pj, Pj, «S:c., be reduced so as to 

 have a common denominator b, then Pj^, PjiJ, &c. , are integers. 



Multiply in each fraction PiPj* &c., numerator and de- 

 nominator by «, taking n such that p-^-^nl, p^^b, &c., are 

 integers. Put nb=x. 



Then of x equally likely cases, VrX is the number favourable 

 to C,.. And, as above, the number of these again favourable 

 to E, \spr^rX, that is, the number favourable to E happening as 

 result of Cr (or " Cr if E")=/'rPr^; and the total favourable 

 to E is 2P/JC. 



Now, if the event E happens, the total of possible cases, of 

 which one or other must be the true one, is clearly IpVx, and 

 by hypothesis none of these cases has any preference over the 

 other, and all are " equally likely ; " while the number of them 

 favourable to E resulting from Cr is pr?rX. Therefore the 

 probability if E happens that it happens from Cr is 



It may be noticed that a proof of the theorem that if A and 

 B are not independent, Prob. A with B = Prob. A x Prob. B 

 if A, which is repeated in edition after edition of ordinary text- 

 books, and so seems to have passed muster, is, nevertheless, 

 erroneous. 



The formula is proved correctly for two independent events, 

 thus : — 



Let a be the number of cases in which the first event may 

 happen, b the number of those in which it fails ; a' the number 

 in which the second may happen, and b' the number in which it 

 fails, the cases for each event severally being supposed equally 

 likely. Each of the {a + b) cases may be associated with each of 

 the (a' + b') ; thus there are {a + b){a' -^-b') compound cases which 

 are equally likely. In aa' of these both events happen ; there- 

 fore the probability of both happening 



Prob. first event x Prob. 



(a + b) {a' + b') a + b a' + b' 

 second event. 



It is then added that the above proof may be applied to two de- 

 pendent events, for we have only to suppose that a' is the number 

 of ways in which after the first event has happened the second 

 will follow, and b' the number of ways in which after the first event 

 has happened the second will not follow. Now if this substitu- 

 tion be made in the above, the first step of the proof will be 

 " each of the [a + b) cases may be associated with each of the 

 (a' + b') cases; thus there are {a + b)]{a' + b') compound cases 

 which are equally likely." But this is impossible. Each of the 

 {a' + b') cases is one in which the first event happens, and there- 

 fore none of them can be associated with any of the b cases, 

 because these presuppose that the first event has not happened. 

 The {a' + b') cases, in fact, can only be associated with the a cases 

 out of the {a + b), and thus the total number of the compound 

 cases intended is not {a' + b') {a + b). A proof can easily be given 

 on the lines already indicated. 



If Pi,/i, n, and b have the sahie meanings as before, the whole 

 number of equally likely cases is nb, the part favourable to the 

 first event is Pi«(5, and the part of these favourable to the second 

 is /iPi«3 (as above shown), which is therefore the number 

 favourable to the concurrence of the two events. The 

 probability, therefore, of the concurrence is 



Certain confusions which often arise in the statement and 

 application of the mathematical theory of probability would be 

 avoided if a clear idea were formed of what is exactly meant by 

 the fraction which is said to represent the probability of an 

 event. 



A good statement of the ordinary account of it is given m 

 Todhunter's Algebra : " If an event may happen in a ways and 

 fail in b ways, and all these ways are equally likely to occur, 



the probability of its happening is — -r, and the probability of 



its failing is This may be regarded as a definition of the 



a + b 

 meaning of the word probability in mathematical works." 

 A definition must not assume and use the notion to be 



defined. Here probability is defined through cases " equally 

 likely to occur " ; but " equally likely to occur " means equally 

 probable, and so the definition assumes the very notion which 

 causes difficulty, the notion of " probability " or likelihood, 

 and of which we require the explanation. 



The first thing to settle is the meaning of these " equally 

 likely " cases. Is the equal likelihood a quality in things them- 

 selves, or is it something in our minds only ? If it is a quality 

 in things it can only mean equal possibility of occurrence or 

 realisation. But if a number of cases, mutually exclusive as 

 intended in the above definition, were in the nature of things 

 equally possible, not one of them could happen. If the claim 

 of any one of them in reality were satisfied, so must the claim 

 of any other, since these claims are equal, and therefore if one 

 happens all must, but by hypothesis if one happens no other 

 can ; thus the only possible alternative is that none of them can 

 happen. (It is precisely on this principle that we decide that 

 the resultant of two equal forces at a point, whose directions 

 include an angle, cannot be in any other direction than the 

 bisector of the angle, and that there can be no resultant of two 

 equal forces which act in opposite directions). 



The equal likelihood then intended cannot be anything in the 

 nature of things because it is assumed that one of the equally 

 likely cases will happen. It is really only in our minds, when 

 there is an equal balance of reasons for and against two or more 

 events, and due solely to our ignorance, since if we knew which 

 was to happen there could be no such balance and indecision. 

 This is clear if we consider what is the reason why we pronounce 

 one event more likely or probable than another ; it is because 

 we think there is more evidence in favour of the one than in 

 favour of the other, however the " more" may happen to be 

 measured. Two events are equally likely to us when we know 

 nothing more in favour of the one than we do of the other — 

 when the state of our knowledge and (it is important to add) of 

 our ignorance, is the same for both contingencies. This view 

 agrees with the actual procedure in mathematical examples. If 

 a bag contains n balls, and one is to be drawn " at random," 

 there are said to be n equally likely cases, that is, each of the n 

 balls is equally likely to be drawn. Clearly this only means 

 that as we don't know how the hand is going into the bag, we 

 have no information in favour of the drawing of any one ball as 

 compared with any other, and no information against the draw- 

 ing of any one ball as compared with any other. 



" Equally likely " cases then being such that owing to our 

 ignorance the evidence in favour of any one is no greater or less 

 than the evidence in favour of any other, the meaning of the 

 definition of probability above criticised is evident ; it is not a 

 definition of probability, but it is the definition of a certain way 

 of measuring evidence. 



We are entitled to say that one event is more probable than 

 another when the evidence before us, being decisive for neither, 

 that in favour of the one seems to us, according to some standard 

 of measurement, greater than the evidence for the other. Now 

 what the mathematical analysis does is not to alter the ordinary 

 meaning of *' probability" at all, but to find a standard for the 

 measurement of the more and less in evidence. 



The whole possibility before us in any given contingency is 

 divided into a number of cases, " equally likely " or "equally 

 possible," in the sense that they are equal from the point of view 

 of the evidence in favour of each of them ; then if one event has 

 more of these equal possibilities in its favour than another, it has 

 in this sense "more" evidence in its favour, and so in accord- 

 ance with the usual meaning (as above described) of "more 

 probable," is more probable than the other. And here the 

 "more or less" in the evidence is not a mere " more or less," 

 but has a definite numerical measure. The evidence being, so 

 to speak, divided into equal units, the strength of the evidence in 

 favour of a contingency is measured by the number of these units 

 in its favour. Thus if the total of equal possibilities, one of 

 which must happen, for the events A B and C is «, of which a 

 involve A, b involve B and c involve C, the comparative strength 

 of the evidence in favour of A, B and C respectively is measured 

 by the ratios a:b:c, while the strength of the evidence of A, B 

 and C respectively, as compared with the evidence for one or 

 vy^her of them happening (which is certain), is, on the same 



principle, measured by the ratios -, - and -. 



n n n 

 If, then, we symbolise the strength of the evidence for A, B 



and C by -, -, and -, and similarly that for one or other 



NO. 1624, VOL. 63] 



