14 



KNOWLEDGE • 



[June 2, 1882. 



(!5ui- iirlatljcmatiral Column. 



INVERSE PROBABILITIES. 



THE formulas we have obtained sliow that in a set of ilrawings 

 or trials of any sort sonic results are more probable than 

 others. For instance, if we took a set of twelve balls out of a bag 

 of l.liOO.OOO balls, of four colours equally divided, we should bo 

 more likely to draw three of each kind contained in the bag, than 

 to draw one of one specified kind, two of another kind, also specified, 

 four of another specified kind, and five of the remaining class, but 

 less likely to draw three of each kind, than one of one kind, two of 

 another, four of another, and five of the remaining class, without 

 specilieation.* And so in a variety of cases we can compare the 

 chances of different results when the antecedent conditions arc 

 known. But it is clear that this power of partial forecasting implies 

 a power of inferring antecedent conditions from observed results. 

 We may now enter upon this interesting department of our subject. 

 In so doing, we are preparing to discuss that application of the laws 

 of probability which is chiefly to be considered in discussing the 

 result.'! of observation and experiment. But it is well to promise 

 that the two departments of the science of probability are most 

 intimately associated together, insomuch that one cannot well bo 

 studied without the other. This will appear, indeed, at tlio very 

 beginning of our discussion of indirect probabilities. 



We know from the laws of direct probability that if there aro in 

 a bag ten balls, three white and seven black, the chance of drawing 



awbiteballis j^ , and the chance of drawing a black ball is j^- 

 Xow if we suppose the ten balls all alike in colour (say all 

 white), but three of them bearing a mark not discernible by tlio 

 drawer (who, however, is supposed to be aware that three are 

 marked) it is clear that when he has drawn a ball, although ho 

 cannot tell whether it is one of the three marked balls, he 

 knows that the probability of his having drawn a marked ball is 



10- 



suppose these ten balls put into a bag with twelve others, 



these others being black; and that a white ball is drawn. In this 

 case, as in the former, the drawer knows that the probability of 



3 

 the white ball being one of the marked three is j^. The addition 



of the twelve black balls diminishes the chance of drawing one of 

 the marked three, but when the fact is known that a white ball has 

 been drawn, the chance that this ball is one of tlie marked three is 

 obviously not a whit affected by the addition of the black balls. 



Xow, suppose that instead of twenty-two balls in one bag there 

 are two bags, each containing eleven balls — in one bag the three 

 marked white balls and eight black ones ; in the other seven white 

 and four black balls. If a bag is to bo selected at random and a 

 ball to be drawn at random from the bag thus selected, it will be 

 obvious that the chance of drawing one of the marked balls is pre- 

 cisely the same in this case as in the former ; for the chance of 

 selec'ting the right bag of eleven balls is precisely the same as the 

 chance that one of the eleven balls now in the bag would be drawn 

 from lie original twenty-two, that is, is one-half. Otherwise, the 

 chance of drawing a marked ball ivould bo affected by separating the 

 twenty-two balls into two sets. For inst ancc, if the three marked balls 

 were put into one bag, and the remaining nineteen into another, it 

 ii clear that the chance of drawing one of the marked balls would 



3 

 be one-half instead of 55. But the two bags containing the same 



namber of balls,t the chance of drawing one of the marked balls is 

 nnchanged. 



Bat we have seen that when a white ball has been drawn from 



• As examples remove much of the seeming mystery of general 

 laws, I will comparr; thcRC three chances together. The chance of 

 any particular result is the same appreciably as though a ball were 

 drawn at random and returned, the operation being repeated twelve 

 times ; and this chance, ag:iin, is exactly the same as though the 

 bag contained only 1 red, 1 white, 1 black, and 1 green ball. So 

 that the p, </, Ac., of onr formula are each erjnnl to unity — the total 

 numlier of trials is 12, and the first probability wc require relates 

 to the drawing of each ball three times. The value of this pro- 

 bability is, theraforc — 



112 



IM'.l'.l' 



(0 



bag), the probability tliat it is one of tho 

 a. white ball has been drawn from 



one of the two bags of eleven balls the chances that this ball is ono 

 of the marked balls — in other words, the chance that it has come 

 from tho bag into which tho mai-ked balls were all placed — is still 



;? 



lij' 



Now the marking ^^•ns only 

 the white balls from II ■' 

 put into separate hi 

 guished by being imi 

 imagine any markiii.^ ; n I i 

 arrived at: — If there ino i\\>' 

 three white balls, the other • 

 balls, then if a bag is solecto 1 

 from this bag is white, the pi i 

 white balls was selected i 



for distinguishing certain of 



I'm riiso where tho balls are 



; I ills aro in effect distin- 



' iliatwe need no longer 



I (lie conclusion we havo 



. 1 laining eight black and 

 lull liliick and seven white 



"I ;i ball taken at random 



hill llio bag containing three 



10 



And by following the method whereby this special result was 

 obtained, it is easily seen that the following general law can be 

 deduced : — If there are in each of two bags p balls in all, q of the 

 balls in one bag being white, and r of those in the other j then if a 

 bag is selected at random and a ball drawn at random from this bag 

 is white, the probability that the bag containing (j white balls was 



selected is — - — Of course, the probability that the other bag 

 r + q 



was selected is 



It will bo observed that p does not appear 



in either result. 



Now take tho case where tho bags do not contain the same 

 number of balls. Suppose one bag contains eight balls, throe of 

 which aro white and five black, and the other tvrelve balls, seven of 

 which are white and five black ; and supposing a white ball drawn, 

 let us inquire what is tho probability that it came from the former 

 bag? 



Hero wo can obviously reduce the problem to tho former case by 

 changing the number o"f balls in the two bags without modifying 

 tho proportion of black and white balls. Thus, taking 24, tho least 

 common multiple of 8 'and 12, wo see that the fijst bag may bo 

 replaced by ono containing 24 balls, of which 9 are white ; whilo 

 tho second bag can bo replaced by one containing 24 balls, of whiol> 

 14 are white. The clianfe of drawing a white ball from one or 

 other bag is in no way modified by these changes, and consequently 

 the inferences to be deduced when a white ball has been drawn aro 

 not modified. But the numbers being now e(|ual, we learn from 

 what was shown in the former case, that if a white ball is drawn 



tho probability is - 

 bag. 



23 



that it 



taken from the first 



|4 



Now, by striking out common factors, it 



(iii.) . 



J3 J3 J3 J3 4' 



I write out I'.l', Ac, in full, to show tho connection between tho 

 I.ririflion and onr formulas.) Now, tho second probability wo 

 luirc relates to the drawing of a definite number of each kind, 



i.tcifying what kind is to be drawn once, twice, Ac. If the chance 



were required that 1 red, 2 white, 4 black, and 5 groon balls would 

 be drawn, the e.tpression for tho probability would be — 

 |lg I'.l'.V.V 



ll \± \± W' 4:" 



And wo should get the same probability, whatever the specifications 

 might be. Further, since there aro 14s_ that is 1.2.3.4 different 

 specifications possible, tho third probability which relates to the 

 drawing of 1, 2, 4, and 5 of different kinds, without specifying 

 which kind is to appear once, twice, four times, and five times, is 

 equal to 



|12 I'.l'.V 



]2_ |£nL ■*■" 



readily seen that the 

 expressions (i), (ii), and (iii) aro to each other as ^ — ^ — jg, 



^ and — : or, as — to —„ to _ ; so that (iii) is the 

 14.4.5 4..'> 108 480 20 



greatest, and (ii) tho least, as was to be shown. 



+ The reader should most carefully nolo tho point of the roa- 

 Honitig hero. If we put an equal number of balls into each bag, 

 wo have not modified £he pi-obability that the ball actually drawn 

 will belong to ono sot or to tho other equal set ; the chances were 

 equal before the separation, and they remain cfiuiil after the sepa- 

 ration. But if wo put into one bag a smaller number of balls than 

 we put into tho other, wo have modified tho clianco that the ball 

 actually drawn will belong to the larger or to tho smaller set; tho 

 chances were not equal before tho separation, but they are equal 

 after it. 



