484 



• KNOWLEDGE 



[Majicu 31, 1882. 



wo hud n loplotnni witli twciily fnccii, iiuiiibcnxl 1. 2, 3, up to 20. 

 Thi'ii till- (iilili- fiiriiipil on the plan iiliown nlMivc wimld liiko nl<in(f 

 timo in llin writing'. Hut it in only noccdKiiry to nutiro linw tlio 

 above tabic in foriui'il, t» ul)tnin ii hinipli' ^ic-mnil rule for nil Rucli 

 onaoa. Wt* Imvf* in h\x KUccoHKi\'U colnnniH tlio ttix nuniliorH-— 

 1, 2, 3 -Ci, rm'li n iM-iitcil nix linioH, in onlcr Hint tliny inny nppcnr 

 in company willi tlu" hjimii' nix nnnilxT'i, 1, '2, a — •>, no tluit tlip total 

 nnmlior of pnim in IJ linu's 0. And, oljviou»ly, if nny numbiT, an ti, 

 roplnord (!, wo iihonlil (-ot 7i linioK ii, or «', nB tlui totnl nnmbor of 

 pair*. In like niunncT, wo hnro in tlio nliovo table, & tiniuH 5 un- 

 favuumblc cnsi'H ; nnd if 7i inatciid of Imd been tlic numbor of 

 |K>8sibl>' rasL'« tiikon 8in(,'ly, w<i Bliimld hnvo had (ii— 1) times (n — l) 

 or (n — 1)' for tlio tot«l numbor of unfavourable canes. Ucnco the 

 chance that anv particular rciiilt of the n would happen onco, at 

 n»-(H-l)' 2ii-l 



least, in two trials is 



n' 



But now to return to our die, wo can extend our inquiry to tho 

 coso of three trials or more. For, in order to obtain tho total 

 number of possibU? sets of 3 rows, wo need only conceive first 1 

 taken with all possililc nets of 2 throws, or 3C times, then 2 taken 

 with all those 30 po.ssible sots of two throws, then 3, then 1 ; and so 

 on. Wo K<"t thu.i, in all, C times 30 sets of 3 throws, or 216 such 

 sets. And in order to liiid all tho sets of 3 throws, not containing 

 1, wo have only to take first 2, then 3, then 1, 5, and 6, with tho 25 

 possiblo sets of two throws in which 1 does not appear at all, and 

 this gives us 5 times 25, or 125. 



. 125 

 Ilonco tho chance that ace will not bo thrown in 3 trials is ,77-, ; 



21b ' 



and tho chance that it will be thrown once at least in thi-ee trials is 

 210- 125 ^j. ^ 



216 " 216' 



And clearly we have this general rule. If there are the same n 

 possible events in each of threp trials, the chance that some 

 particular event will not occur in any of the three trials is 



i— ^-— , and the chance that it will occur once at least in one of tho 

 n' 



three trials is 5 ^ . And by proceeding in this way wo get 



the yet more general rule : — If there are the same n possible events 

 in each of r trials, the chance that some particular event will not 



occur in any of tho r trials is ; and the chance that it will 



occur once at least in the r trials is " ~^ ~ ' . 

 n' 



This general result is of extreme importance, as we shall presently 

 see. It is of importance, not only in inferring the antecedent ]iro- 

 bability, as to the result of successive trials, where th9 conditions 

 of each trial are known, but also in inferring from the results of 

 successive trials (or observations or e.vperiments, if we please) the 

 conditions (supposed unknown) under which those trials have been 

 made. 



I close this paper with two simple examples of the application of 

 this rule. 



1. What IS Ike least nuviber 0/ trials which wo«id give a person at 

 least an equal chance 0/ throwing ace icith a single die ? 



We have seen that his chance of failing to throw an ace in two 

 . 25 125 



trials la gjj, and in throe trials ;jw;. In four trials the chance of 



625 " 125 



failing will bo T^T^. Now we note that ^Tg is greater than a half, 



625 

 and ,.,„g is less than a half. Hence in three trials he is more 



likely to fail than to succeed, and in four trials he is more likely to 

 succeed than to fail. Therefore four is the number of trials re- 

 quired. In any continued scries of sets of three trials, he would 

 fail somewhat oftenor than he would succeed, in throwing one ace 

 a^ least. Hut in any continued series of sets of four trials, ho would 

 Buccccd oftenor than ho would fail. 



2. What is the chance that, in three iossings 0/ a coin, head will 

 appear once <il I'asI t 



In each trial there are two possible events, i.e., the n of our rule 

 is e(|unl to 2. Thus tho chance that liead will not be tossed in three 



1> I 

 trials — or -. Therefore the chance that head will api)uar once 



at least in throo 'ossings is -. The odds are 7 to 1 that one head 



at least will bo tossed in three trials ; and if there were to be re- 

 peateil sots of three trials, a bettor backing the appearance of one 

 head at least in each set should Iny thesB odds. I-'urther, if a 

 person is to receive JCS in case a head a^ipears in three trials, he 

 ought to pay £7 for his chance. 



MATIIKMATICAL QOEBIES. 



[39] — ClIANXEH. — Ki^quired — 



1. The chunco of dealer (at Whist) holding only one hononr in 

 any pnilicular suit. 



2. The chunco of 1 he dealer holding at least one honour in any 

 particular suit. 



3. Tho chance of tho dealer holding only one hononr ia 

 Trumps. 



•1. The chance of tho dealer holding ut least one honour ia' 

 Trumps. GRADATUf. 



[" Uradatim " sends solutions of these problems. — Ed.] 



[10] — What is tlie general solution of the equation 

 <m / (19 \. 



^, + «(^}+6e=o 



and show'how it ia obtained. — WlLOELH. 



AXSWEES TO QUERIES. 



[27] — The value of a diamond varies as the sqtiare 0/ the weigh^^ 

 A diamond is broken into three parts : determine the probahle volw 

 0/ the parts, compared with tlutt 0/ the unbroken diamond. m 



Lot X, y, and a — (x + y) be tho weights of the parts into wbuS 

 the diamond is broken, a being tho weight of the whole diamond 

 and .'. .r'-l- i/' + (a — x-l- !/)* = Bam of value of parts. To find th* 

 mean average value, aasumo that x remains constant, and that Jf 

 has every value between and (a- 1) ; add up the value of th0 

 above expression ; and divide by tho number of values. That 

 is, integrate the above expression, with reference to g, from to 

 (t-x, and divide by a—x. Hence we get 



., , (n-xY (a-x)' 



3r' + 2(a-x)» 



a—x 



Give to X every possible value from to a, and divide by tli^ 

 number of values. That is, integrate the above expression from 

 to a, and divide by a. Hence, we get — 

 , 2a' 



3 5 , 



3a 9 

 Ilcuce, the required proportion is— T. B. 



Hornek's Method. — In reference to the method of extractmg 

 "All Roots," given in "Our Alathematical Column" at p. 4M, 

 allow me ta observe that it was not discovered by '" the late 

 Mr. Homer, M.P.," but by William George Horner, of Bath. I 

 have a special personal interest in mentioning this ; for on tlw 

 l)ublication of my "Practical Arithmetic for Senior Classes," in 

 1^58, the Athenwum, while commending the introduction of Homei'B 

 method in the " Extraction of Roots," pointed out that I had mad* 

 tho very mistake of referring to Francis Horner, which your coire- 

 spoudent now does. On writing to Bath, I got the full name ap 

 above, and 1 made the necessary corrections in the second edition.'^ 

 Henry G. C. Smith. \ 



0m- ©Leftist Columiu 



rilUE following game, from the Westminster papers for Auguat^ 

 JL 187-1-, is selected as illustrating the M-eakuess of lead from 

 short suit ; even when the odd trick only has to be made to win, and 

 the idea is not to bring in long suit, but to use long trump suit to 

 ruCf. Our correspondent, Mr. Lewis, was Y. His lead at trick 7 

 is worth noticing. 



A. The U.\nds. 1'. 



Siiades — Kn, 6, 5, •!, 2. 

 Diamonds — 5, -1. 

 Clubs--Q, 8, 3. 

 Hearts— A, Q, 10. 



B. 

 /S'jKirfes — A. 



Diamonds— A, K, 10, 3. 

 Clubs— K, 9, 7, 2. 

 Hearts— Kn, 7, 0, 3. 



Sixtdes—Q, S, 7, 3. 

 Diamonds — Q, Kn. 

 Clubs— A, Kn, 6, $, 4. 

 Hearts — 5, 2. 



Z. 



Spades— K, 10, 9. 

 Diamonds -9,8,7,6,8. 

 Clubs— 10. 

 Hearts— K, 9,8,4. 



Score.— ^ B, i ; Y Z, 4. 



