October 1, 1887.] 



♦ KNO^A^LEDGE ♦ 



277 



event. (About three-quarters of a million of trials would 

 give an even chance.) But if an experiment is commenced, 

 and after long and wearisome series of trials — say, few 

 thousand — no such series should appear, as is exceedingly 

 likely, then the experimenter would argue that, since by the 

 laws of probability the unsuccessful trials already made can 

 have no eflPect on the results of future trials, we are no 

 nearer success than at the beginning. Another series of 

 10,000 trials might be carried out with similar want of 

 success, and another, and yet another — a hundred such 

 trials taking us to a million : if all these were similarly un- 

 successful, and there is no reason why they should not be, 

 we should be as far from success as at the beginning. Going 

 on in that way would be useless — it would seem — or, in 

 other words, the laws of probability have in this case mis- 

 led us. 



The fallacy of all this lies in the assumption that that 

 which is likely is sure to come off. If we fail in 10,000 

 trials we may equally fail in the next, and the next, and so 

 on. But inasmuch as we have about one chance in 100 

 of succeeding in 10,r00 trials, whereas at each trial we have 

 but about one chance in a million, we must not treat a 

 chance of the former sort as if it were no greater than a 

 chance of the latter. And to bring the matter down to the 

 test of manageable experiment — it is almo.st certain that in 

 10,000 trials there will occur one case at least of thirteen 

 successive " heads " — this has been tried, apart from the 

 numerical considerations which show it to be certain. 



Imagine, then, 80 trials of this kind, giving almost cer- 

 tainly about SO runs of 13 heads (for if some gave none, 

 these failures would be made up for by others which gave 

 two or more such runs). Here, then, are 80 cases of 

 13 heads in succession to which we may limit our attention, 

 for no shorter runs can have any bearing on our attempt to 

 get 20 successive heads. We may regard the continuance 

 of these 80 trials — to see how many more heads there may 

 be — as if they were fresh trials. Now it would not be 

 thought a very wonderful thing if in 80 trials we obtained 

 a run of seven heads, which with the thirteen already given 

 would be the 20 heads required. Hence this reasonmg 

 based on actual experiment shows thit in 800,000 trials 

 there is a good chance of tossing 20 heads in succession, 

 wonderful though such a series may be, and incredible 

 though it might seem during actual experiment on so 

 wearisome a scale that any good could result from going . 

 on with the tossLngs. 



It must be remembered in all such cases that success may 

 come off at the first trial, and is quite as likely to come off 

 at that trial as at any other. It is this circumstance which 

 causes the answer to the famous Petersburg problem to be 

 so seemingly monstrovis : — How much should be paid, asks 

 that problem, for the privilege of tossing a coin and receiv- 

 ing -L, Al., 81., 161., or so on, always doubling, according fus 

 you fail to toss head at the first, second, third, fourth, &c., 

 trial. The true answer is that no sum, however large, will 

 fairly represent the price you should pay. You have but 

 one trial, just as in a lottery you may have but one ticket ; 

 but the number of prizes is infinite, and you mat/ draw one 

 of the biggest in your one trial. According to the laws of 

 fair lotteries, you must pay for your chance of getting each 

 ticket. Thus you must pay '21. for the chance — equal to 

 certaint}' — of getting the smallest prize at least ; you must 

 pay half of il., or '21., for your chance (one-half) of getting 

 the il. prize ; you must pay one-quarter of 8/., or, again, 

 2/., for your chance (one-quarter) of getting the 8/. prize ; 

 and so on, always the same sum of 21., for your chance of 

 getting the 16Z. prize, the 32^. prize, the 64/., l'2Sl., and so 

 on for ever. Hence you ought to pay, if you should have 

 such wealth, an infinite number of sums of '21. This sounds 



like nonsense, because if you have unlimited wealth it would 

 be far better to keep it. Yet the answer is right, regarded 

 merely in its numerical aspect ; for the sums you have the 

 chance of winning are enormously greater, one may say 

 infinitely greater, than the sum you are risking, let it be 

 what it may. 



The lesson really taught by the Petersbui-g problem is the 

 folly of the gambler's notion that his chance of winning a 

 stake represents its actual value to him. We see at once 

 that a man must be an idiot wiio would pay, say, 10,000?. 

 for a chance in such a lottery as is described above, though 

 the actual value of his chance is very much greater. But a 

 man of the gambling type does not hesitate for a moment 

 to regard a ticket out of 200,000, where the prize is 

 200,000/., as worth not only II., its mathematical value, but 

 much more — he will gladly pay three or four pounds for 

 such a ticket, which, regarded as property, is worth much 

 less even than the \l. which represents its due proportion of 

 the great prize. 



The readiness, too, with which many will buy tickets in a 

 lottery, and their unwillingness to buy a chance in any such 

 venture as the Petersburg problem, shows how little men 

 realise the meaning of large numbers. A gambler begins to 

 see what he is doing when tossing is in progress, the first 

 "tail" (let us suppose) to limit the amount of his prize. 

 But when he hears of a scheme like the Louisiana Lottery, 

 in which there are a hundred thousand tickets, he buys his 

 ticket with the vaguest idea as to what such a number 

 really represents. If it could be pictured to him in a suit- 

 able diagi'am, or if he could be shown a heap of grain in 

 number equal to the number of tickets, and recognised one 

 only among all these as representing his chance, he would 

 probably think twice before investing his money on so pre- 

 carious a venture. But as matters are, a certain air of 

 mystery hangs over the event : something occult seems in- 

 volved ; in such sort that superstition comes in to suggest 

 that by noting omens or dreams, or by venturing on some 

 specially selected number, he will have a better chance of 

 winning than that mere 100,000th, or one- thousandth, or 

 whatever it may be, which the laws of probability assign to 

 him. 



There are other points in regard to which the ways of the 

 gambling community are interesting, not to say amusing 

 and unreasonable. The regular gambling fraternity, those 

 who make a business of it, are apt to express contempt for 

 the way in which the verdant gamblers whom they despoil 

 aim always at getting a great prize for a small risk — as in 

 buying a lottery ticket, backing the favourite, and so forth. 

 The}' themselves, however, are quite as unwise in their 

 systematic gambling, and would lose in the long run quite 

 as much if they limited all their ventures to fair ones. One 

 can lose a fortune, for example, quite as readily by always 

 laying the odds as by always taking them, if the true odds 

 are equally followed. It is only because laying the odds 

 gives the bookmaker a better opportunity to bet on unfair 

 terms that in the long run he always comes off best, while, in 

 the long run, the backers of favourites are brought invariably 

 to ruin. But the bookmaker falls into the mistake of sup- 

 posing that in the system of laying odds against events more 

 or less unlikely to happen — and generally even the favourite 

 horse in a race is more likely to be beaten than to win, 

 though more likely to win than any other individual horse 

 in the race — there is safety, and not only safety, but the 

 assurance of success and wealth in the long run. 



It is for this reason, probably, that while we find the 

 inexperienced pigeon seeking always some such way to wealth 

 as the purchase of a lucky ticket in some great lottery, or 

 backing hor.ses at long odds and winning, the more prac- 

 tised gambler of the rook species is the victim of a delusion 



