April 21, 1882.] 



KNOWLEDGE 



541 



And by proceeding in this "way it will be found that (on the 

 Hsnmption made), if thirty-two persons engaged in the spoeulation, 

 ■D average of £7 wonld have to be paid to each ; if sixty-four 

 engaged, an average of £8 to each; it 128 engaged, an average of 

 £9 to each ; if 256, £10 each ; if 512, £11 each, and so on. The 

 general rule being that, according to the assumption, if 2" jjcrsons 

 engaged, an average of ii + £2 would have to be paid to each. So 

 tbat as there is assumed to be no limit to the number of persons who 

 maj try their chances — or, what practically comes to the same 

 thing, no limit to the number of trials which may be made — we have 

 • 88 large as we please, and therefore (n -t- 2) £ the average number 

 of pounds the bank would have to pay for each out of 2" trials may 

 be made as large as we please, or equals infinity. 



It wiU be noticed that in the above remarks I have not overrated 

 the value of the several chances. For instance, if there are eight 

 tossings, the occuiTence of four tails cannot be thought an unlikely 

 erent. And in one respect I have systematically underrated the 

 TSlne of each set of trials ; for when but one person is left who has 

 not tossed " head," I have invariably supposed the single toss to give 

 "head." It is easily seen that the effect of this is to diminish the 

 estimated value. In fact, two trials where four persons are engaged 

 oorrespond to one trial with eight persons ; four trials with four 

 persons ' correspond to two trials with eight, or to one trial with 

 sixteen persons; so that, as might be expected, the repetition of 

 any of the several kinds of trial above considered leads to a steady 

 increase (on the assumption made thronghout) in the mean value of 

 each person's expectation. 



It is also well to notice how slowly this mean value increases 

 with the increase of the number of trials when once we have 

 leached large numbers. Thus, for 2,018 persons, the mean value 

 of each person's expectation is £13, and for 1,096 persons, the 

 mean value is £14; an increase of only £1. though 2,018 persons 

 are addeil ; and 4,096 persons must be added to increase the mean 

 value to £15 ; 8,192 persons more to increase the mean value to £16 ; 

 and so on. 



But now, returning to our million of tossers, let us consider how 

 their various fortunes illustrate the general doctrine of probabili- 

 tiee, and more particularly the subject of luck. When we consider 

 the million as a whole, we find nothing in the result of the tossings 

 irhich seems to indicate either good or bad luck ; for iu each fresh 

 aeries of trials about one-half have tossed "■ head" and about one- 

 half " tail." But it we conceive the various individuals of our 

 army of tossers to remain unaware of the real nature of the process 

 in which they are taking part, and only to know the results of a 

 few tossings taking place in their immediate neighbourhood, it will 

 be seen that opinions resembling those formed in the world at largo 

 respecting good luck and bad luck would be found among our 

 tossers. Those 210, or thereabouts, who tossed " tail " twelve times 

 nmning, would be regarded by those around them (severally) as 

 exceptionally lucky men. Many might be disposed to back the luck 

 of one or more of these fortunate individuals of whose success 

 they might become cognisant. These 210 are not a whit more 

 likely (severally) to toss "tail" than to toss "head" at the 

 eleventh tossing ; and yet if one were to reason with those who 

 backed one of the lucky 240, it might be found very difficult to 

 persuade him of the folly of his course. One might reason that 

 there was no such thing as trustworthy luck ; that though such and 

 anch a tosser had been lucky so far, yet no inference could be drawn 

 from his past success as to the success of his next venture ; and so 

 OD. But the reasoning would seem good in answer, that there must 

 be such a thing as good luck, for had not this particular tosser 

 thrown " tail " twelve times running, whereas no one else of those 

 aroimd had thrown " tail " more thun four or five times running ? 

 His luck had been trustworthy in the past, why might it not be 

 trusted as respects the future also ? In fine, the proposing backer 

 might remain obstinate in the belief that he was doing a rather 

 clever thing in backing the luck of the fortunate tosser, and perhaps 

 at heavy odds. 



On the other hand, there is a line of reasoning equally unsound, 

 by which a directly opposite conclusion may be reached. A person 

 who had heard of the tossing of " tail " twelve times running, might 

 Oonclnde that '" head " would be almost certain to come at the next 

 trial. We can see that this is not so, when we remember how our 

 840 (or so) successful losers are to proceed to a thirteenth trial, and 

 that only about half of them may be expected to succeed. But any 

 reasoning founded on the abstract probabilities might fail in this 

 case, as in the former ; because specious reasoning may bo urged in 

 favour of failure on a thirteenth trial. Thns it might be urged that 

 to toss " tail " twelve times running is altogether unusual ; much 

 more, therefore, must it be unusual to toss " tail " thirteen times 

 nmning. And the reasoner, forgetting altogether that the only 

 question he has to consider is the single tossing about to take place, 

 and its chances, might confine his attention to the d priori impro- 

 bability of tossing thirteen "tails" in succeesion. In betting on 



the result, he might persuade himself that it was this unusual event 

 he was betting upon, and so take heavy odds against it ; whereas, 

 in reality, the event he was betting upon would bo simply the result 

 of tho tossing of a coin once. It is certain, at any rate, simple as 

 the question is in reality, that nine men out of ten do reason in this 

 unsound manner.* 



Two highly important lessons may be drawn fi'om the considera- 

 tion of these matters, and it wonld bo well if those who have a taste 

 for gambling would study those lessons carefully. 



In the first place, we hear accounts from time to time of very 

 lucky gamblers ; of rtms of luck by which men have " broken tho 

 bank " at Baden or Homburg, and so on ; and many are led to 

 believe that there really is such a thing as luck that can be depended 

 upon, and so are encouraged either to court fortune by backing 

 those who have been lucky, or else to try whether thoy may not 

 themselves be lucky in gambling ventures. The consideration of 

 the St. Petersburg problem has shown that where many gamble, 

 there must be some who have an extraordinary run of luck. 

 Because, although the problem as dealt with only relates to the 

 tossing of a coin, it is obvious that similar conclusions would have 

 been deduced, whatever ventures had been considered, and even 

 though the odds wore heavy against success in each separate 

 venture, instead of being even, as in the case of tossing a 

 coin. If a large number of men cast each a die, about a 

 sixth will throw Ace ; of this sixth, again about a sixth 

 will throw Ace on a second trial, and so on ; and clearly, 

 it only requires that the original numbers should be large 

 enough, to get several who will throw ace, ten, twelve, twenty, or any 

 number of times running. And in evcrv such instance we shall 

 always have our lucky men, amongst whose ranks, however, the 

 next trial will make the same relative gap as among a similar 

 number of untried, or of hitherto unlucky, persons. So it is with 

 the multiplied trials continually going on in the gambling world. 

 There must be many seemingly lucky men ; and there must be some 

 few who seem lucky, even among the lucky. But neither the lucky, 

 nor the luckiest of "the lucky, arc better worth backing in a new 

 venture than some unfortunate who has hitherto never had the 

 smallest modicum of good fortune. Take a man who has broken 

 the bank half-a-dozen times at Baden or Homburg, and let him 

 risk his money on some fair venture with a man who has never sat 

 at the gambling table but to lose every penny in his possession, yet 

 there is not a straw of odds upon either. 



The other lesson is equally important, and the mistake which it 

 tends to correct has been as mischievous in its results as the one 

 just considered. The belief that " the luck must change " has over 

 and over again led the unfortunate gambler to persist in making 

 fresh ventures. "I have been unfortunate so Icng," he reasons, 

 " that now I may expect a run of good luck ; to give up gambling 

 now would be to throw away the good fortune I have been so long 

 waiting for." The Petersburg problem teaches precisely the same 

 lesson respecting Ul-fortune as respecting good fortune, since the 

 same results wonld follow whether we regarded the tossing of 

 " tail " as an event to be rewarded by a money payment, or 

 as an event which should compel the loser to pay money. 

 We see that the sequence of many events of the same kind 

 — i.e., a run of luck — can teach us nothinj as to future 

 events. A run of bad luck should be regarded by the 

 gambler as belonging altogether to the past ; the " whirligig 

 of time " may or may not " bring in its revenges," or what may 

 appear as such ; but the past ill-luck of the gambler will in no sort 

 affect his future fortune. He has not the slightest valid reason for 

 expecting a run of good luck to counterbalance his former bad 

 luck.t 



(To be continued.) 



* The old story of the sailor, who put his head through a hole 

 made by a ball in the side of his ship, confiding in the improbability 

 that a second would strike the ship in the same place, is true to 

 nature ; — only we are not bound to believe that the saUor was a 

 Briton. 



t A change of luck he may, in one sense, expect ; that is, he may 

 hope not to have a run of bad luck snch as he has already had. But 

 he has no other reason for hoping this than the actu.il improbability 

 of a run of luck, either good or bad, in a given series of trials. A 

 man who has lost five games (of pare chance) in succession, may 

 expect a change of luck, in so far as he may hope to win some, at 

 least, of the next five games. But he has no better chance of 

 winning some of these five games than he would have had if the 

 first five had not been played. Thus a cessation of bad luck re- 

 peatedly takes place when many games are played. If the seeming 

 change of fortune follow after a change of seat, or the use of a new 

 pack of cards, or some like observance of gambling superstition, the 

 fact is noted (the failure of the observance would not be noted) and 

 tho superstition is encouraged. 



