>2G 



• KNOWLEDGE • 



[April 14, 1883. 



^ur iVlatbrmatiral Coliimtu 



P K () B A B 1 L I T I E S. 



By Tim EiiiTok. 



LET 11" next iloftl witli iiomo cii»os not altogether no ninijilc iis 

 tliono hilhi>rto ronHiilrrcd. 



SupiKiiio thnt tliprc is n lottery, in which there arc more prized 

 t han one, the priie» being uncqunl in value — how can wo determino 

 the value uf a ticket ? 



Take n simple bat definite CMO : — 



Suppfof thfre art ten tiektlt, all equally Ulely to b» drawn, and 

 that thrre are three prizes, worth reepectivety £6, £3, and £1, what 

 it the lvalue nj a finale ticket ? 



Tlie prizes are together worth £10. It follows that the ten 

 tickets must together be worth the three prizes together, for any 

 one buying all the tickets would get all the prizes and no more. 

 Ilence the ten tickets must together be wortli £10, or (since they 

 arc all equal in value) must be worth £1, just the same price as 

 when there is a single prize of £10. 



And manifestly this is so in every case. It matters not how the 

 prizes are distriliuted, the value of one among n tickets is always 

 one nth part of the total value of the prizes. 



It is probably this simplicity in lotteries of this kind, and the 

 consequent obvious nature of the fraud when the total value of the 

 prizes is less than the total amount received for tickets, which has 

 cau8c<l those who have taken advantage of the weakness of human 

 nature for gambling, to adopt] various systems in which the swindling 

 is as great, or greater, but is not quite so obvious. All the existing 

 lottery systems, and all the gambling games carried on formerly at 

 such places as Ilombnrg and Baden, and now at San Jlarco, are so 

 arranged that the luck may for a while run against the lottery 

 holders or the "bankers" at roulette, rouge et noir, and the rest. 

 The swindlers who thus encourage gambling can truly say that 

 they take their chance of loss, and even of serious loss. They do 

 lose at times, heavily ; but in the long run they always come out 

 right, the percentage of profit, estimated from mathematical con- 

 siderations, is invariably attained. 



Nay, these gambling rascals not only adopt systems by which 

 they may occasionally lose, but they affect to allow privileges by 

 which, as it seems to the inexperienced, they must lose. They allow 

 that very system of wagering to which we referred some time since 

 OS one by which, to all appearance, one player must always win, — 

 the system of doubling the stake after each loss until finally a win 

 leaves a balance of gain as against several previous losses. 



Let us take the simple case stated by ns before, and see where 

 the fallacy about sure gain lies : — 



A tcsset a coin \rilh B, staking £1 ; if he loses he stakes £2; if he 

 loses he stakes £1; and so on, doubling each time until he wins, when 

 he clears £1. And, as he must at length win, he can keep o» adding 

 pound to pound, ad infinitum, yet each separate wager is fair. Where 

 is th» fallacy ? 



The fallacy resides in the supposition that A must at length win. 

 He may go on doubling till he no longer possesses enough money to 

 wager again on this doubling system, or till, having wagered more 

 than he possessed, he is unable to pay. He is then ruined, and the 

 process of adding pound to pound comes, perforce, to an end. In 

 the long run, if B only has money enough as compared with A, this 

 untoward event is bound to happen. If A and B have nearly equal 

 capitals at starting the case is in some degree different : A may ruin 

 B. But in the case of the bank at San Marco, or wherever else 

 gambling may be pursued (of course mere coin-tossing is not the 

 method but only illustrates the method), there are multitudes of 

 small -■<'» all risking their "small peculiar" against the possessions of 

 one big (and very busy) B. They are inevitably absorbed separately 

 if they are only possessed strongly enough by the gambling spirit." 



Ix>t us see what are the respective positions of A and B with 

 regard to the prizes actually at stake in this case :— 



At each venture A plays for £1, and we may consider that B 

 stakes £1. If A'a capital allows him to go on doubling ten times 

 before he gives in, B plays for what A will have to i)ay him if he, 

 A, is obliged to stop. The amount will bo the sum of A's succes- 

 sive payments up to, and inclusive of, the tenth doubling, or 

 £l-i-£2 + £l + £,S+ .... -H £256 + .£51:.' = £1,023. 



The sum of the prizes is therefore £1,021; and there are 1,02 1 

 possible events, for there are 2 jMiseiblo events at each tossing, and, 

 therefore, 2'° jiossible events in 10 tossings. Therefore, at each 

 Tontore (not at each tossing, but at each setting-off upon a series of 

 tossings with constantly doubled stakes) B is practically in the 

 position of one holding a ticket in a lottery of 1,024 tickets, each 



price<l at £1, and a single prize of £1,021 [for note that we must 

 not call the prize £1,023, any more than in tho case of a fair 

 lottery of 10 £1 tickets, we should coll the prize £9, becauitc 

 that is all tho winner really gains, £1 having been paid for 

 his ticket]. Or wo may say that B is in the position of one 

 who pays £1 for the chance of drawing one particular ball out of a 

 liiig of 1,021', £1,021 having to bo paid him if he is successful. We 

 know that in tho long run— in a few millions of trials, for instano- 

 — ho would draw successfully about one 1,021th the total number of 

 trials. We know further that there would be times when he would 

 be behind, and times when ho would bo ahead of this average. 

 The times when he was ahead would be bad times for A. If B"* 

 capital enabled him to continue long enough he would be practicallT 

 certain to ruin his oi)ponent. 



In their excessive desire to swindle the people, GovcrBment." 

 which have encouraged lotteries have tried to devise the most 

 attractive forms of wagering, submitting to mathematicians the due 

 discussion of the probability problems involved. One of the most 

 remarkable occasions of this kind on record is that which gave rise 

 to what is called the Petersburg problem. It occurred to the 

 Russian Government to start a lottery on the following plan : — 



Each person who took part in it was to venture the same fixed 

 sum of money £x, on the following conditions : — A coin is to be 

 tossed until bead appears ; if head comes at the first toss the- 

 person is to receive £2 ; if at the second toss the person is to re- 

 ceive £4; if at tho third toss, he is to receive £8; if at the fourth 

 £1C, and so on. The difiicnlty was to determine at what amount 

 should X be fixed P 



The answer given by mathematicians was not encouraging. We 

 can form a tolerably clear notion of the sum we should care to risk 

 on such a venture ; and I suppose no one would be inclined to place 

 that sum very high. If a lottery were actually est.iblished on such 

 a principle as the above, and £10 or £12 were set as the value each 

 person was to pay for his venture, very little business would be 

 done at the price. But mathematicians asserted that if the value 

 of X were set at any sum, however large, the " bank " would 

 inevitably lose in the long run, supposing only that a sufficient 

 number of ventures were made on such terms. For example, say 

 the value assigned to x were £1,000, then, although it would be 

 madness to risk such a sum on a single trial, yet if many millions 

 of ventures were made, the bank would be immensely in arrears 

 when a balance came to be drawn on the results of all the trials. 

 The theoretical value of x is, in fact, inyijii(y. 



This curious paradox is described by Professor De Morgan as 

 affording one of the most instructive lessons on the subject of 

 probabilities. 



I shall explain hereafter the reasoning by which the above 

 seemingly paradoxical, but undoubtedly true, answer is obtained. 



PROBLE.M. — Can the following be solved by elementary geometry ? 

 In a triangle given, a + b, a + c, and the angle A to construct the 

 triangle ? 



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