I(i0 



KNOWLEDGE 



[March 24, 1882. 



©ur i¥latl)rmntiral Column. 



1' U(tR A II I 1,1 T I E8. 

 By tmk KniTOR. 



LET Ufl ncxl tnko n case not <|uil(> «o simple ns tlio todHin)? of n 

 coin, niiinoly, the caatinK of a die. We know tlint tlic cliftncc 

 of throwing nee in h single trial is }. Let us consider what is tlio 

 cliance of throwinR nco in two trials with one die. The considera- 

 tion of this will brinp before us a very common mistake in denling 

 with chtineo questions, nn instance of whirli occurred recently in 

 the discussion of the question about cutting any one of three 

 named cards (of any suit) from n pack, onco in three trials. A 

 correspondent asked, with reference to this question, whether, as 

 the chance of cutting one of the cards in a single trial was 

 obviously f\, the chance in three trials must not be thrice this, 

 or V'j' This is erroneous, but not very obviously so. So again 

 in the case of a die— it is not obvious at first sight that the 

 following reasoning is unsound. The chance of throwing aco in 

 <mc trial is ,',, therefore in two trials it must be J, or^. \Vo sec, 

 however, at onco that the following reasoning is incorrect, or, at 

 least, leads to an incorrect result, though it is precisely the same in 

 character. Since the chance of tossing a head in one trial is i, the 

 chance of tossing a head in two trials must be j, or certainty ; for 

 we know there is no certainty at all in the matter. Yet even here 

 it is not quite clear to the beginner where the error comes in, and 

 he is often inclined to think there must be some defect in our 

 method of representing chances, when reasoning which seems 

 correct leads to an obviously incorrect result. 



Kow the answer usually made to the above incorrect reasoning 

 about the tossing of a die, runs commonly as follows : — The chance 

 that an ace will be thrown in the first trial is i, but the chance that 

 an ace will be thrown in the second trial is not ^, because there may 

 be no second trial, for the first may give an ace. We must, there- 

 fore, add i, the chance in the first trial, to J (the chance in the 

 second trial), reduced in a degree corresponding to the chance that 

 a second trial will be required. Now the chance that there will be 

 a second trial is, in fact, the chance that the first trial will fail to 

 give an ace, or J, so that the chance of throwing ace in a second trial 

 8 not jl, but only five-sixths of i, or -^. Adding this to ^, the 

 chance of throwing ace in the first trial, we get ^ -I- -^^, or |i, for the 

 chance of throwing an ace in two trials.* 



But the objection suggests itself to the student that the second 

 trial may be guaranteed, whatever the result of the first trial. The 

 thrower may say — to begin with — I mean to throw this die twice ; 

 what is the chance that one of the throws at least will bn an ace ? — 

 and then the above reasoning about the contingent nature of the 

 second throw is rendered unmeaning. De Morgan deals with this 

 objection in a very just way, but I am not sure that his reasoning 

 convinces all minds very readih-. Todhunter, after noting the 

 objection, says, " The error really arises from neglect of the follow- 

 ing consideration : when events are mutually exclusive, so that the 

 supposition that one takes place is not incompatible with the sup- 

 position that the other takes place, then, and not otheru-ise, the 

 chance of one or other of the events is the sum of the chances of 

 the separate events. In the present case success in the first trial 

 is not incompatible with success in the second trial, and therefore 

 we cannot take the sum of the chances as the chance of success in 

 one or other of tlie trials." Hut this, after all, amounts only to a 

 statement of the fact that that reasoning is erroneous by which the 

 chance of throwing ace in two trials with a single die is made to be 

 t^vice J. Now, this fact we knou.; because we see that the extension 

 of the same principle of reasoning leads to an obviously incorrect 

 result. What we want is to learn exactly where the error lies. I 

 do not find that this is clearly sho>vn in treatises on probability. 



Let us take an illustrative case from which, as I judge, the true 

 nature of the error may be learned. 



In an urn there are six balls, marked from 1 to 6. The chance 

 of drawing ball 1 is, of course, the same as the chance of 

 throwing ace .at a given trial with a single die ; that is, it is -J-. 

 Now suppose that six persons draw each a ball. One of them must 

 have drawn ball 1. The chance that any one of the six has drawn 

 this ball is J ; and the chance that one of a given pair of these six 

 persons has drawn the ball is J + ,\. This is clearly the case, as 

 shown in paper I. ; and that the reasoning is just is proved by the 



• What follows is quoted, with very little change, from a series 

 of articles on the " Laws of Chance as Applied to Statistics," which 

 I wrote eleven years ago for the Knglish Mechanic, where they ap- 

 peared in August and September, 1871. I shall take occasion, 

 when convenient, to borrow passages from those articles, but with 

 such moditicationB as my experience of the dilKculties commonly 

 found by students of the subject may suggest. 



fact that when it is oztondod so on to include all the Biz porsona, wa 

 got six times ^, or unity, corresponding to the certainty that one of 

 the six has drawn ball 1. Now the fallacy in the former reasoning 

 aliout the die lies in the supposition that twa throws with a single 

 die give the same chance of throwing an ace that any pair of our 

 six ball-drawers has of drawing ball 1. Whereas it is obvious that 

 to roijresent the case of the die-throwing, we must have — not two 

 different bulls drawn at random from an urn containing six, but one 

 ball drawn at random and replaced, and then again one ball drawn 

 at random. 



Let it lie noted that there is no begging of the question here. 

 It is certain that the chance of throwing an ace is the same as the 

 chance of drawing ball 1 from the urn containing six. It U 

 certain that to represent the second throw, as well as the first, the 

 urn must have its full complement of six — that is, it is certain the 

 ball first drawn must be rc|>laced before the second drawing ia 

 made. Whereas it is certain that the case which gives as the re- 

 sulting chance i+ i, is the case where a ball is drawn, and then (or 

 simultaneously, it matters not which) another ball. 



That the two cases arc distinct is rendered obvious, therefore. 

 And not onlj- so, but we can see which case gives the better chance. 

 For in considering the two cases, we can place our 6nger on the 

 exact spot where the chances differ. Suppose that a person A 

 proceeds as in the former case, a person B as in the latter, each 

 dealing with a separate urn, containing balls numbered from 1 to 6; 

 and let us compare their chances of drawing ball 1. They begin 

 alike. A draws a ball from his um, and B one from his. Their 

 chances of succeeding in this first drawing are, of course, equal : bnt 

 if they fail, their chances on the second drawing are not equal. For A 

 has to return the ball he drew into the um again ; and he will have 

 no better chance of success at the second trial than at the first. 

 But B retains tlio ball first drawn, and at the second trial he has a 

 better chance of success than at the first ; for he has to draw now 

 ball 1 from an urn containing only five bails instead of six. But B's 

 chance in his drawings i.^ certainly ^ -*- ^ ; A's chance, therefore, is 

 certainly something less than ^ + -^-. 



We see, then, that we must adopt a more trustworthy mode of 

 reasoning in the case of successive trials under unchanged con' 

 ditions. 



A Pretty Geometbic.^l Problem, axd Mogul's Pboblek. — A 

 great number of solutions of these problems have been received, 

 and of the former (" Kelland's ") problem a very complete discus- 

 sion has been sent to us. It will be a work of some little time to 

 analyse all the solutions, but we hope next week to give an abstract 

 of " Mogul's " solution, and of the paper just mentioned, with 

 suitable figures. Both problems are very instructive. — Ed. 



^ur Cftrss €oInmn. 



How the Devil was caught. Played at Brighton, 1S79. 



AUgaier Oamiit. 



Warrs. Slice. Whttk. Buck. 



Mephisto. F. Edmonds. Mephisto. F. Edmonds. 



1. P to K4 



2. V to KB4 



3. Kt to KB3 



4. P to KR4 



5. Kt to Kt5 



6. B to B4('>) 



7. B takes QP B to Kt2 



8. P to Q3 P to KB3 



9. Kt to K6 B takes Kt 



10. B takes BC-") P to B6(') 



11. P takes P(') QtoQ3 



P to K4 

 P takes P 

 P to KKt4 

 P to Ko 

 Kt to KK3C) 

 P to Q4('^) 



12. B takes P(b) Kt takes B 



13. P takes Kt Q to Kt6(ch) 



14. K to Bsq Kt to B3 



15. Kt to B3(') Castles QRC) 



16. Kt to Q5 R takes Kt(i) 



17. P takes R Kt to Q5 ! 



18. P to B3 P to KB4 



19. P to Kt5 R to KsqC') 



20. P takes Kt B takes P 



21. Q to K2(i) R takes Q 



22. K takes R Q to Kt7(ch) 

 I resigns. 



(*) Not to be commended. Black only obtains a very indifferent 

 game by this move, whereas, by the usual continuation of 5 P to 

 KR3 I he ought to get the better game, in spite of White's subse- 

 <iuent attack. If a player is afraid to expose himself to the attack, 

 then the more logical course would be to refuse the Gambit from 

 the beginning. 



C") 6 P to Q4 is the proper move here, for, if Black plavs 6 P to 

 KB3, then 7 B takes BP, 7 P takes Kt, 8 (B takes P and White 

 wins his piece back, but we usually prefer 8) P takes P. as this 

 sacrifice yields some interesting play. 



(') Whereas, now he might have played P to KB3 and won the 

 Knight with tolerable safety. 



