'11 J. 



KNOWLEDGE 



[March 10, 1882. 



ThPKC ronrontioni ore in ronlity nil which tho iitudont of proba- 

 bilitioii hn<i to mnko ; nil others which soom to ariic, n« his stu'ly 

 of thp suhji'ct pruccc'ilJ, ore in renlily only noco-Hary con«LM|iicni<>g 

 of those. 



Lot us now consider sonip simple illnstrations of the rennlts of 

 these conventions. 



Sii|>|i(i5<o n lin(f contains ten Inlls, nil white. Then the chnncc 

 (lint II white liiill will bo dniwti JM iiiiitv, and the chance that a black 

 bull will be driiwn is 0. 



If III! the ten balls in a bag lire of the ("nnie size, nnd the drawer 

 ran n« readily reach one im another, or if tho ba^ is well shnki'n, so 

 that each ball has an eijiial chance of coming to tho top, the chance 

 of any gi^'cn ball beiii); dniwn at a given trial must bo exactly equal 

 to the cluince of any other. All the balls are equally likely to be 

 ilrnwii. So that if C bo the clianci' for each, 10 C is tho chance for 

 nil the ton. But one of them n.ust be drawn. Hence IOC must be 



e inal to certainty, i.e. to 1 ; nnd C" — 



. ^^ . 



This is obvious enough, but let us submit it to a test. 



Suppose one ball is white and the remaining nine balls all black, 

 and Buppo.so that if the white ball is drawn, a certain prize, say 

 i'lO, will bo won. Now, if there are ton pcrsoiis to draw, nnd each 

 takes one ball, it is certain that one of them will draw the white 

 ball. But there is nothing to prevent one and the same person 

 baying np the chances of all the ten, or the certaii.ty of drawing 

 tho white ball, and so winning the prize of £10. lie should clearly 

 pay the same snm to each, fir each has the same chance of 

 winni:ig, and the total sum he shoulil expend should be clearly £10, 

 for he makes certain of getting that sum by buying up nil their 

 chances. Since, then, he has to pay £10 in equal sums to ten 

 pers ins, he must give £1 to each. This, therefore, is the value of 

 eich person's chance of drawing the white ball, and this sum is one- 

 tenth of the sum at stake. This is tho same as saying that the 

 chance of each is -^. 



Wo have here spoken of a bag of 10 balls ; but it is clear the 

 reasoning ajiplies to any case in which there are a number of events 

 all equailj- likelv, and one of which must happen. If there arc 



1 

 13 balU in a bag, the chance of drawing any particular ball is rr/ 



1 

 If there are n, the chance of drawing any given ball is -. If a die 



has all its si-t faces numbered ditfercntiv, the chance that when the 



1 . . . 



ihe is thrown any given number will be uppermost, is tt, if the die is 



not loaded, — that is, if it as likely tliat one face will be uppermost 

 us another. Again, if a coin is tossed with a good high spin, and 

 the coin is uniform, so as to spin with perfect freedom, the chance 



of head or tail is -. So the chance of drawing any particular card 

 from a picquet pack of cards i-s — ; the chance of drawing any par- 

 ticular card from a whist pack is — ; and so with other such cases. 

 52 

 Next, suppose that a bag contains three white and seven black 

 balls, or ten in all, and let as inquire what is the chance of drawing 

 a white ball. 



seem, then, sufficient to reason that, since there are three white 



1 



ing any one of tho three is — . The reasoning is, indeed, just, and 



tho result is correct, but tho student cannot be too careful in avoid- 

 ing all hasty conclusions in these questions of chance. We shall 

 see presently that a line of reasoning which seems at first sight 

 quite as just as the above will lead to an obviously incorrect result. 

 It will be well, then, were it merely to initiate n system of close 

 inquiry into these matters, to discuss tho alwve result somewhat 

 nttentively. We shall save time in the long run by getting these 

 seemingly axiomatic nir.ttcrs thoroughly reasoned out. 



.\pplying onr former method of measuring chances by consider- 

 ing the value of the right to draw tickets in n lottery, we readily 

 determine the chance we are seeking. We have only to suppose 

 that there are three prizes equal in value — say each £1. Then tho 

 chance of each of tlie ten drawers must needs be equal. Now, 

 anyone who bought up all these ten equal chances should clearly 

 pay £3, since this is what ho would obtain when tho ten tickets were 

 drawn. Each ticket would thereforo cost six shillings — that is, the 



value of tho chance of each drawer is rr: of Uie prizo gained by a 



■nccouful drawing. Wc infer tho justice of the conclusion thatth* 

 chanco of drawing ono white ball fr«jn> a hog containing seven black 



3 

 nnd three white balls is T^y 



Since the above reasoning is npp'iculde, whotevcr the total 

 nninber of balls, and whatever the numlier of white balls, wo bar* 

 this general result, that the chanco of drawing a white ball from • 



r 

 bag containing n balls, of which r are white, is -. 



Further, since tho reasoning is as np]>licable to the black balli M 

 to the white, it is obvious that the chance of drawing a black ball 



out of a bag containing seven black and three white balls ''tj;' 



This ii the chance of /ui'.inij to draw a white ball. And, generally, 

 the chance of failing to draw a white ball from a bag containing • 



balls, of which r are white, is ~ . 



Nor is the reasoning affected if the balls which arc not white ore 

 of more than one colour ; while the same reasoning applies to the 

 balls of divers colours. So that wc clearly get this general mle, 

 including all that wo have thus far attained to: -If there are H 

 balls (all equal in size) in a bag, of which v are white, I black, 

 r red, ;; green, and so on, till all the colonrs and balls in the bag ara 

 reckoned, then — 



The chance of drawin 



v.hitc ball is 



failing to draw a white ball is 

 drawing a black ball is 



failing to draw a black ball is - 



drawing a i-ed ball is 



,, failing to draw a red ball is... — - 



n 



and so on through all the colours, and we may also combine any 

 of the colours together in such statements as the following: — 

 The chance of di-awing either a white or a black ^ 6 + ir 



ball is ) ~, 



The chance of failing to draw either a white or i n—{b + tv) 



a black ball ) JJ 



and so on 



This general law is applicable to any case where the chance is 

 required that one of a certain set of events will happen out of a 

 larger number of events whose chances are equal. Thus, the 

 chance that either ace or deuce will be thro^vn with a single die 



2 

 is- 



The chance that a court card will be drawii out of a picquet 

 12 



pack of cards is 



32 



the corresponding chance in the case of a 



whist pack being -— . 

 52 



[Solutions of problems, by T. R. and others, in our next. — Ed.] 



A Substitute Foa Water ix Foot-W.\kmess. — People who 

 travel much in winter, either in railway carriages or in any of the 

 other modes of conveyance, are continually annoyed and incon- 

 venienced by the fact that the hot water in their tins gets cold very 

 soon ; in fact, if tho tins are to be of any comfort to the traveller, 

 they must bo changed every two hours. Who, amongst all those 

 who havo felt starved and miserable through the water in the foot- 

 wanner having become cold, will not hail with delight the fact that 

 the science of chemistry promises speedily to bring a fresh boon 

 to the traveller in all parts of the world, in the form of a foot- 

 w;iriiier whicli will keep hot for a period of ten hours, at the same 

 time giving out four times as much nsiful heat as water ? The name 

 of the chemist who is bringing about this gi-ent and useful change 

 is M. .\ngolin, who purposes using crystallised sodium acetate 

 instea 1 of water. When once the tins are filled, the stoppers well 

 soUlertd, and the warmer perfectly air-tight, all trouble ceases 

 o.Tcopt wai-ming them up when required for use. Tho tins can be 

 used over and over again, the salt being perfectly stable. Experi- 

 ments are now being tried on tho London nnd North-Wcstem 

 Railwav, also on various foreign railwavs, with a view to its adop- 

 tion.— F.C.S. 



I 



