440 



♦ KNOWLEDGE ♦ 



[March 17, 1882. 



refoiTod to tlicno oniiily-unilonitood illustnitiniiR. Thun (to take in 

 pnsiinga vcrj' fmnilinr raiip), Buppono wo nro told thnt tho bottinK 

 IH four to om- nKii""' " forlniii liorBO in n riico, wlmt nro wo to 

 infer i» tin- viilui> of tlir i-lmnco wliicli \iottin(f csiMTtB nHHlfOi to 

 liim. Till" oJil» of four to ono nirjin, of cunr»o, tlnit tlioro nro four 

 cliancCA npiiniit llio liorao for ono in liis fnvonr, or tlint out of Hvo 

 oqutti cliunci'8 lio lui« ono. Uin clinncc of winning ifl, tlicri'foro, an 

 far (U the liettin',1 shniis, oquni to that of dniwinn ono piirtiiMilnr 

 ticket from ii liu|? .-ontnininK live, nil ociunlly likely, untocodently, 

 to be (Irnwn. 1 need liardly Hny tlmt the reiil ehnnce of the borne 

 nniv be ven,- difTcrcnt. 'I'lie betting exi^rta niny not know neiirly 

 so inucli as they suppose, nnd tho horse niny have n niucli better or 

 a much worse ehnnce thun they iniBKino. Or n^ain, they may know 

 ((.omo of them) much more than they pretend to know. JJut so far 

 as tho l)etting shows, tlmt is the horse's chance. We shall see 

 further on how tliis sinii)lilication of the jirublem enables us to 

 determine from tho known hotting about two horses singly, the 

 priiper betting about tho pair, and similarly for throo or more 

 horses. 



Again, tnko cases which at first sight do not seem to resemble 

 the problem wo have just considered— as, for instance, the tossing 

 of a coin or the casting of a die more than once — and let us see 

 how those can be reduced to tho simpler case. Here is a simple 

 problem of this kind : — 



What is the vhance that, when a coin is tossed twice, the tosaings 

 u-ill beunlike — (hat >'«, that loth mil not be heads nor hoth tails? 

 Simple as this problem is, by 'the way, tho great mathematician 

 d'Alembert went astray in dealing with it (at a time when the 

 mathematics of probabilities were not very well understood). Ho 

 reduced it to our general law in summary but inexact fashion. Thus 

 he said, There are three possible events: either both to.ssings 

 will be' heads, or both tails, or [they will be unlike; therefore 

 the chance that they will be unlike is one in three, or 

 one-third. This result d'Alembert maintained witli a degree of 

 confidence which seems singular when the simple nature of 

 his mistake is considered. To solve the problem correctly, we 

 must i)roceed, as he did, to consider the various possible events ; 

 but not err as he did, by failing to notice that he counted two of 

 them as one. The possible events are four, viz., head, head; tail, 

 tail ; head, tail ; and tail, head. There are manifestly no others, 

 and as manifestly any one of these is as likely as any other. Now 

 of these four cases, "two give unlike tossings, viz., the two last. 

 The real chance, therefore, is not one in three, but two in four, or 

 one half. 



If d'Alembert had been a betting man, and had backed his 

 opinion bv wagering two to one against the occurrence of unlike 

 headings," during a great immbtr of trials, lie would have lost 

 heavily — the real wager being even. In the long run, half the 

 trials weuld be of the kind against which he had wagered ; and 

 losing thus as often as he won, while paying twice as much when 

 he lost as he received when he won, he would manifestly lose just 

 as much as his opponent had wagered. At least, the result would 

 approach this (and his loss, therefore, be the greater) the more 

 numerous the trials. 



We note, then, in passing, that men who are unwise enough to 

 gamble ought to have a clear idea of the laws of chance ; for in the 

 long run these are as inexorable as the laws of the Medes and 

 Persians. A man may buy a single ticket in a lottery, and he may 

 chance to win, or he may try his luck pretty often at roulette or 

 rouge el noir, and still be a winner ; but if he keeps on long enough, 

 he will inevitably lose in proportion to tho calculated chances 

 against him. (Ho may lose at the outset, and probably, on the 

 whole, it is the better for him that he should.) 



There are, indeed, ways in which men suppo.'ie that with patience 

 they can always win, though slowly. The following is a familiar 

 illustration, which wo leave as an exercise for the reader : — 



Two gamblers, A and B, toss a coin on tho following terms. A 

 wagers against B evenly £1 ; if B wins A wagers £2 even with him ; 

 if B wins again A wagers £4 even with him ; and so on, doubling 

 each time, till A wins the toss. When this happens, whatever the 

 number of tossings before the event, \ wins £1. They begin again, 

 A wagering £1 even, as before, and doubling till ho wins, when he 

 pockets another £1. Every wager is .strictly fair, yet every trial 

 (as A must, at last, win) ends by A. winning £1. The system seems 

 altogether unfair in its results, though perfectly fair in details. Is 

 it Bo or not ? Assume A and B to have equal capitals, say £1,000, 

 and estimate their chances of success or failure. .\t first sight it 

 seems as though A must gradually win every pound B has. In 

 reality it is not so, as wo shall show later. 



" All Roots." — (See " .\nswers to Correspondents," p. 391, line 

 17.) In reply to " All Roots' " query, I give an easy method of 

 extracting tho fifth root of 5153(532, and will endeavom- to state the 

 rule as briefly as ]ii)s.«ible. Make as many columns ns the number 



of tho root to bo extracted ; then, having found the first quotient 

 figure, put it in tho first column ; odd tho same numlior to it ; then 

 multiply tho sum obtained, placing tho product in tho second 

 column. Continue this course, viz. : add and multiply, add ami 

 multiply, decreiiHing one column i-nch time, till you have only one 

 to make in tho first column, which will Ijo tho same nundwr of 

 additions as the root to be extmeted. Then odd to the first 

 <oliiinn, 00 to the second, 01)0 to the third, 0000 to the fourth, and 

 bring ilown the next period in tho fifth. Then find tho probable 

 number of times that the fourth column will be contained in tho 

 fifth ; put this in the qnotient. and also under the first column, nnd 

 nild and multi|)ly as before. This method was discoverod by the 

 late Mr. IIonier.'.M.I'., ami is applicable to all roots. Your corre- 

 spondent will find all jiarticiilurs in " Kavunagh's Arithnvtic." — 



HoMEO. 



Find tho fifth root of 51 53032. 



5153632 (22 

 2 4 8 It; 32 



XoTiCE. — The necessities of space oblige us to defer solutions of 

 problems till next week. — Ed. 



(Buv WBf)i^t Column. 



By " Five of Clubs." 



THK PENULTIMATE. 

 Sir, — I send you a game for young players, illustrative of the 

 above heading. Playing the penultimate is, leading the lowest but 

 ono in suits of more than four cards, which are not headed by the 

 Ace, or do not contain two commanding honours or strong sequences, 

 <ic. This mode of play is, practically, the invention of Cavendish, 

 and is the logical extension of the lead of the lowest but one, in 

 suits containing intermediate sequences. I look upon the invention 

 as little inferior, in its means of conveying information (and its 

 consequent influence upon the game), to the convention which 

 requires a player to return his lowest in suits of four cards, or his 

 highest in a suit of tliree. In point of fact, in some cases it conveys 

 more rapid information, for the lead from five cards may be, and 

 often is, declared in the second round of the suit. And where tho 

 dealer, being happily possessed of five trumps, is forced before 

 trumps are led, and tnimps with the penultimate, the information 

 of a minimum of five trumps is conveyed at once. 1 ought, perhaps, 

 to state that, unless there are special reasons to the contrary, the 

 lowest but one should still be led in the trump suit, notwithstanding 

 that it might be headed by Ace, King, or Queen, with small cards. 

 It is time that, occasionally, a weak lead simulates a lead from five 

 or more ; for a player, unknown to his partner, may have his strong 

 suit attacked by the adversaries. In such cases, whore the lead is 

 not an original lead, a jiartner must draw his inferences with more 

 caution. Pkeliekick H. Lewis. 



The H.4X[.s 



Cluls—i, 3, 2. 

 Hearts— K, Kn. 1, 

 Spades— Kn, 8. 3. 

 Diamonds — 8, 0. 



B. 

 Ctubs—Kn,10,9, 8. 

 Hearts— A, Q,0, 6. 

 Spades — A, 5, 4. 

 Diamonds — K, 4. 



Score .—Three all. 



Clubs — 7, 6, 5. 

 Hearts — 10, 7. 

 Spades — 10, 7, fi. 

 Diamonds— A, Q, Kn, 



Clubs- A, K, Q. 

 Hearts— 8, 5. 

 Spades— K, Q, 0, 2. 

 Diamonds— 10,9,7, 2. 



