578 



KNOWLEDGE <» 



[May 5, U 



(But iiflatbrinatiral Column. 



FAIR HUT UNWISE BETTING. 

 By the Editok. 



BEFORE considering other nnicrs of chance problems, it imU bo 

 well to consider the relation between tho mnthematioal c\ mce 

 of an event and the moral value of expectations depondin),' m n it. 

 For convenience, lot us do this with special reference to wager;- upon 

 eventa raoro or less probable, such as races, matches, and so Inrth. 



If the chance of an event is -, the chance of its failing to happen 

 n 



is . Comparing these two chances, wo get the ratio r to n — r, 



in which r represents the number of favourable cases, and n—r the 

 number of unfavourable cases. The technical expression used to 

 indicate this relation is that the odds are r ton — r on the event 

 (that is, in favour of it), if r is greater than n — r; orn — rtor 

 against the event, if r is less than n — r. 



Suppose now that in an nm there are ten balls, of which three 

 are white and seven black; then the chance of drawing a white 



ball is — , and the chance of failing to draw a white ball is wViIln 



10 * 10' ^""^ 



the odds against drawing a white ball are 7 to 3. And if two 

 persons. A and B, were to wager on the event, A to win if a white 

 ball were drawn, and B to win if a black ball were drawn, then, 

 that the wager should be strictly fair, the sums respectively wagered 

 by A and B should be in the proportion of 3 to 7. It will be clear 

 that this proportion is fair, if we remember the real fact as respects 

 wagers, that when once a wager has been laid, even though the 

 betters keep the wagered sums in their pockets till the issue is de- 

 cided, the case is precisely the same as though those sums were 

 added together to form the prize for the winner. In the present 

 ease, supposing A to wager £3 against £7 of B's, the prize for the 



winner is £10 ; and as A's chance is — , the price he should pay for 



it is three-tenths of £10 — that is, £.3 — while B's price for his chance 

 should be seven-tenths of the prize, or £7. 



Bnt there is another way of \-iewing the matter. Suppose A and 

 B to go on betting upon the same event, A always backing the 

 white and B the black, the drawn ball being returned after each 

 wager had been decided ; then, in the long run, the number of 

 times that A and B would be respectively successful would be in 

 the proportion of 3 to 7, as nearly as possible — the more nearly the 

 longer the backing continued ; and it is clear that, to equalise their 

 chances, the money gained by A and B respectively, when suc- 

 cessful, must be in the proportion of 7 to 3. 



Here, then, we have the mathematical principle on which all 

 wagers should be Imsed, if they are to be fair, — viz., that the sums 

 respectively staked by the bettors must be proportioned to their 

 respective chances of success. 



But although bets made on this principle are strictly fair as 

 between the parties to the wager, yet it is a mistake to conclude 

 that a man's chances of loss or of gain are equal, when he stakes 

 his money on fair wagers. 



For, in the first place, his property is not increased in the same 

 proportion if he wins an even wager, as it is diminished if he loses. 

 Thus, suppose his property to be £1,000, and that he wagers £500 

 again.st £500, the chances of success and failure being equal. If he 

 loses, his property is halved ; but it is not doubled if he wins ; and 

 in like manner it may be shown that, whatever ho stakes, the effect 

 of success is not equivalent to the effect of failure. 



It might seem, however, that if a person always wagered a sum 

 bearing a very small proportion to the property he has at first, he 

 would be safe from serious loss in the long run. Supposing, for 

 example, that a person. A, has £1,000, and repeatedly wagers £1 

 against £1 on equal t<u-ms, it might seem as though he wouid never 

 be much richer or much poorer than at starting. Now, even if this 

 were so, it would be an argument against betting, since it would 

 show the uselessness of fair wagering. But, as a matter of fact, 

 a belief in the " long run " is one of the most fatal delusions which 

 a bettor can entertain. It may be shown— and, indeed, will be found 

 to follow from the principles to be enunciated in these papers — that 

 the chance of absolute ruin, in such a case as wo have imagined, 

 increases with the number of wagers. The ivido of money lost to 

 money won in such a series of wagers approaches, indeed, more 

 and more nearly to equality the greater the number of wagers ; but 

 the extent of the difference between the two sums is likely to 

 bo greater the longer the process of wagering is continued. Thus, 

 in a hundred wagers there would be nothing very wonderful if A 

 lost or won as many as fifty-five wagers, in which case he would 

 have lost or won £10 ; whereas in a million wagers it would be 



utterly improbable that ho would lose or win so many as 550,000 

 wagers ; the numbers of won and lost wagers would probably be 

 much closer; but it would be unlikely that they would bo so clow 

 as 500,500 and 495*, 500; yet if they were no closer, and the halanet 

 were against A., his £1,000 would be lost, and his wagering pnt an 

 end to. It is calculable that the odds are greatly in favour of the 

 numbers not being so close as 500,500, and 400,500, and it 

 is obvious that the balance is as likely to be against A as 

 in his favour. So that what ho in effect would risk by 

 entering on so long a series of wagers would be this, that in M 

 jjrohahility his whole jiroperty would bo as if risked on a single 

 contingency, in which the chance of success or failure was but one* 

 half. No one would think of risking his whole fortune on the ton 

 of a halfpenny ; nor would any one care to agree that his whole 

 fortune should be thns risked, if in drawing a ball out of a, bag of 

 twenty, of which but one was white, he failed to draw the white 

 ball. Yet a person who makes a series of small wagers, trusting 

 to the " long run," is no whit better circumstanced (if he only con- 

 tinues wagering long enough) than one who has agreed to so daring 

 a venture as the latter ; while the longer his wagering is to last, the 

 more nearly does his case approach that of the former. For the 

 complete investigation of the subject of wagering, I would refer the 

 reader to the chapter on the '' Eisks of Loss and Gain " in De 

 Morgan's admirable, though somewhat dry, treatise on probabilities; 

 but the following general principles may be enunciated, as contain- 

 ing the essence of the whole matter : — Better small wagers and 

 many than large wagers and few; better few small wagers than 

 many small wagers ; better yet, no wagers at all. 



[30] — T. F. asks for the solution of equations — 



a:'-f y = ll (i) 



y' + x^ 7 (ii) 

 The equations, of course, reduce to a biquadratic in x or y, having 

 one obvious root, and so reducing to a cubic, the solution of which 

 does not belong to elementary algebra. The four roots are all 

 real, as is indeed obvious if we consider that (i) and (ii) are 

 equations to two parabolas having their axes at right angles, and 

 intersecting in four points. In the only sense in which the equa- 

 tions can be regarded as suitable for our "young readers" as we 

 wrote, their solution is very easy, bece"cp. x = 3 and y = 2 are 

 obvious solutions, so that .7-3 will be a factor ot n^ biquadratic in 

 a-, or J/ — 2 a factor of the biquadratic in )/, according to the line 

 followed in obtaining an equation with one unknown. Or we may 

 write (i) and (ii) thus — 



a-^ -I- V = 3' -H 2 and ir-f t = 2'-i-3 

 whence obviously x = 3 and y = 2. 



The MrscuLAR Pokce of a Crocodile's Jaw. — A strange kiud of 

 experiment has been lately made in Paris by Drs. Regnard and 

 Blanchard, viz., the measurement of the power exerted by the 

 masseter muscle in a crocodile (a muscle passing from the cheek- 

 bone to the lower jaw). Ten live crocodiles of the species C. 

 galeatus or siamensie, that had been sent in large cases from Saigon 

 to M. Paul Bert, afforded the opportunity for such experiments. 

 Some of these animals were as much as 10 ft. in length, and weighed 

 about 1541b. The difficulty of managing such creatures in the 

 laboratory was, of course, considerable. The crocodile was fixed 

 with ropes on a heavy table ; the lower jaw kept in contact with 

 the table by a cord, while the np])er was raised by means of a 

 cord attached at the extremity, and passing up to a beam overhead* 

 A dynamometer was inserted in this cord, and was affected 

 when the animal was stimulated with an electric current. 

 In this way a crocodile of about 1201b. weight gave an indi- 

 cation of about 3081b. (140 kilogrammes). The application of the 

 cord at the end of the suout was necessary, but unfavovirable, 

 seeing the application of the force is thus at the end of a long 

 lever, and there is at least five times more space between this 

 point and the insertion of the masseter muscle than between the 

 latter and the joint of the jaw, the fulcrum. Hence the masseter 

 really produces a force five times that indicated by the dynamo- 

 meter, or about 1,540 lb. (700 kilogrammes). This extraordinary 

 force, it should be remembered, was that of an animal somewhat 

 weakened and at a low temperature. The force (of about 308 lb.) 

 is really applied at the end of four large teeth that project beyond 

 all the others, and considering the surface here represented, the 

 authors estimate the pressure, while the bite is executed by the 

 extremity of those teeth, at nearly 100 atmospheres. Making 

 similar experiments with an ordinary sporting dog. they obtained 

 in the dynamometer a pressure of about 72 lb. ; while the effect at 

 the insertion of the masseter was about 3601b. The pressure at 

 the point of the canine teeth would be about 100 atmospheres. It 

 is calculated that the crocodile is about one-third stronger than a 

 dog of the same weight would lie. — Ttme.". 



