30 



♦ KNO^A^LEDGE ♦ 



[Decembee 1, 1886. 



bottom remaining numbers in column >[. When this pro- 

 cess has continued until all the numbers in column m are 

 scored out, A will be found to have won 10?. ; an<l whatever 

 the sum he had set himself to win in the first instance, so 

 long as it lies well within the tolerably wide limits allowed 

 by the bank, A will always win ju^t this sum in each 

 operation. 



Let us take a few illustrative cases, for in these matters 

 an absti'act description can never be so clear as the account 

 of some actual case. 



Cionsider, tlien, the accompanying account by A of one of 

 these little ojjerations. The amount which A sets out to 



win is, as before, 10/. He divides this 



up into three parts — 31., 31., and 41. 

 He starts with a loss of '2!., which he 

 sets in columns m and l. He stakes 

 next .5/. and loses, setting down .5/. in 

 columns m and i.. He stakes 8/., the 

 sum of the top and bottom numbers in 

 column M, and wins. He therefore sets 

 8/. under w, and scoi-es out 3/. and 51., 

 the top and bottom numbers in column m. 

 (The reader should here score out these 

 numbers in pencil.) The top and bottom 

 numbers now remaining are 31. and 21. 

 Therefore A stakes now iSl. Say he 

 *"^°' ^' loses. He therefore sets down 51. both 



in column M and column l, and stakes 8/., the sum 

 of the top and bottom numbers under m. Say he 

 loses again. He, therefore, puts down 8/. under columns 

 Ji and L, and stakes 11/., the sum of the top and bottom 

 numbers under m. Say he wins. He puts down 11/. under 

 w, and scores out the 3/. left at the top and the 8/. left at 

 the bottom of the column under Ji. (This the reader should 

 do in pencil.) He then stakes 9/., the sum of the top and 

 bottom numbers (4/. and 51. respectively) left under m. 

 Say he wins again. He then puts down 9/. under w, and 

 scores out the il. left at the top and the 51. left at the 

 bottom of the column under m. There now remains only 

 one number under 3i, namely, 2L, and therefore A stakes 21. 

 Let us suppose that he loses. He puts down 21. under m 

 and L, and, following the simple rule, stakes il. Say he wins. 

 He then puts down il. under w, and scores out 2/. and 2/., 

 the only two remaining numbers under sr. A, there- 

 fore, now closes his little account, finding himself the 

 winner of 8/., 11/., 91., and 4/., or 32/. in all, and the loser 

 of 2/., 51., 51., SI., and 2/., or 22/. in all, the balance in his 

 favour being 10/., the sum he set forth to win. 



It seems obvious that the repetition of such a process as 

 this, any convenient number of times at each sitting, must 

 result in putting into A's pocket a considerable number of 

 the sums of money dealt with at each trial. In fact, it 

 seems at a first view that here is a means of obtaining 

 untold wealt.h, or at least of ruining any number of 

 gambling banks. 



Again, at a first view, this method seems in all respects 

 an immense improvement on the simpler one. For whereas 

 in the latter only a small sum can be gained at each trial, 

 while the sum staked increases after each failure in geome- 

 trical progression, in this second method (though it is equally 

 a gambling superstition) a large sum may be gained at each 

 trial, and the stakes only increase in arithmetical progression 

 in each series of failures. 



The comparison between the two plans comes out best 

 when we take the sum to be won undivided, when also the 

 system is simpler; and, further, the fallacy which under- 

 lies this, like evert/ system for gaining money with csr- 

 tainty, is more readily detected, when we consider it 

 thus. 



Take, then, the sum of 10/., and suppose 5/. the first 

 loss, after which take two losses, one gain, one loss, and 

 two gains. The table will be drawn 

 up then as shown — with the balance 

 of 10/., according to the fatal success 

 of this system. 



On the other hand, take the other 

 and simpler method, where we double 

 the original stake after each failure. 

 Then supposing the losses and gains 

 to follow in the same succession as in 

 the case just considered, note that the 

 first gain closes the cycle. The table 

 has the following simple form (count- 

 ing three losses to begin with) : — 



We see then at once the advantage in the simj^ler phin 

 which counterbalances the chief disadvantage mentioned 

 above. This disadvantage, the rapid in- 

 crease of the sum staked, is undoubtedly 

 serious ; but, on the other hand, there is the 

 important advantage that at the first suc- 

 cess the sum originally staked is won ; 

 whereas, according to the other plan, every 

 failure puts a step between the player and 

 final success. It can readily be shown that 

 this disadvantage in the less simple plan just balances the 

 disadvantage in the simpler plan. 



But now let us more particularly consider the pro- 

 babilities for and against the player involved in the plan 

 we are dealing with. 



Note in the first place that the player works down the 

 column under m from the top and bottom, taking oflf two 

 figures at each success, and each figure adding one figure 

 at the bottom after each failure. To get then the number 

 of figures scored out we must double the number of 

 successes ; to get the number added we take simply the 

 number of failures, and the total number of sums under ll 

 is therefore the original number set under Ji, increased by 

 the number of failures. He will therefore wipe out, as it 

 were, the whole column, so soon as twice the number of 

 successes either equals or exceeds by one the number of 

 failures (including the first which starts the cycle). Mani- 

 festly the former sum will equal the latter, when the last 

 win removes two numbers under jr. and will exceed the 

 latter by one when the last win removes only one number 

 under ji. 



Underlying, then, the belief that this method is a certain 

 way of increasing the gambler's store, there is the assump- 

 tion that in the long i-un twice the number of successes will 

 equal the number of failures, together with the number of 

 sums originally placed under Jl, or with this number 

 increased by unity. And this belief is sound ; for accord- 

 ing to the doctrine of pi'obabilities, the number of successes 

 — if the chances are originally equal — will in the long run 

 difler from the number of failures by a number which, 

 though it may perchance be great in itself, will certainly 

 be very small compared with the total number of trials. 

 So that twice the number of successes will difler very 

 little relatively from firice the number of failures, when 

 both numbers are large ; and all that is required for our 

 gambler's success is that twice the number of successes 

 should equal once the number of failures, together with a 

 small number, viz. the number of sums originally set under 

 M, or this number increased by unity. So that Jve may say 

 the gambler is practically certain to Jvin in the long run in 

 any given trial. 



In this respect the method we are now considering 

 resembles the gambling superstition before examined. In 

 that case also the gambler is sure to win in the long run, as 



