September 1, 1887.] 



♦ KNOW^LEDGE ♦ 



253 



and in offering only 1,000 to 1 Lord Yarborough was acting 

 (unintentionally no doubt) like those judicious but dishonest 

 bookmakers who offer the young verdants of the turf 10 to 1 

 against a horse, when the true odds are nearer 20 to 1 — 

 or otherwise shorten the true odds against the horses in a 

 race, making themselves safe on every race, though one or 

 two of their victims win largely for the nonce, and so more 

 surely become their prey, and induce others to fall into the 

 same net. If we imagine 18,280 wagers on the Yarborough 

 plan, about ten would result in loss to the peer ; and the 

 winners, as they pocketed severally their thousand pounds, 

 would be apt to think " What a fool his lordship must be 1 " 

 while bystanders witnessing the transaction would pity his 

 folly, and repeat to themselves the good old proverb which 

 indicates the swift separation of the unwise and his property. 

 But the loss of ten thousand pounds or so (for, of course, 

 there might be two or three more Yarborough?, or two or 

 three less, than the exact average) would be more than 

 compensated, so far as the hereditary legislator was con- 

 cerned, by the sum of 18,270Z. which he would h.ave 

 pocketed during the proceedings. He would probably have 

 secured a profit of 8,270/. ; he would be practically certain 

 to have secured a profit of at least 4,270/., and his risk of 

 not securing the full average profit of 8,270/. would be 

 balanced by his chance of securing three or four thousands 

 more than that amount. 



Such are the conditions and chances in the matter of the 

 notorious Yarborough wagers, which in these days of puritj' 

 are no longer possible — our tempters of fortune having 

 found more speedy ways of ruining themselves in " cover " 

 speculation and the like. 



Xow consider the conditions and chances in regard to a 

 " perfect innings " at cricket. 



Here we no longer have definite chances to deal with. 

 Yet, if we consider a given cricketer playing against a given 

 eleven, it mu.st be admitted by all who know anything of 

 cricket, even with all the glorious chances of the game, 

 that a fair estimate ain be formed of the chance that he will 

 get into double figures. In the first place, his batting average 

 for previous seasons is known. How far during the actual 

 .season he is playing up to his usual standard is also known. 

 The strength of the bowlers and fielders playing against him 

 can in like manner be estimated, and the state of the ground 

 and of the weather are also given conditions of the problem. 

 A good cricketer in such circumstances would not hesitate 

 long in forming an opinion as to the chance that such and 

 such a player going in under such and such conditions to 

 meet such and such an eleven would reach double figures. 

 For instance, if Mr. Grace, or Mr. A. G. Steel, or either of 

 the Reads— amateur or professional — were going to the 

 wicket early in a game against a first-cla.ss county eleven, 

 we know that any cricketer would regard the chance of the 

 player making double figures, the ground being in good 

 condition and the weather favom-able, as considerably 

 more than one-half, more even than two-thirds or three- 

 quarters. On the other hand, when a player remarkable 

 rather as a bowler or a wicket-keeper than as a bat goes in, 

 late in the game, against the same eleven, we know that the 

 chance of his making double figures is much le.ss than one- 

 half, and may even be less than one-tenth or one-twentieth. 

 Of course, the event in either case may upset all such 

 anticipations. A Grace may be out at the first ball, and a 

 Shaw or a Sherwin, the pride of his eleven because of mar- 

 vellous skill with the ball or over the bails, may go in last 

 and pile up thirty or forty runs, to the disgust of the 

 opposing eleven, which had regarded the innings of their 

 opponents as already over. But the estimated chance was 

 none the less correct. 



Now it so chances that just as there are thirteen chances 



to be considered in the case of a Yarborough hand at whist 

 one chance for each card, so are there thirteen chances to be 

 considered in the case of a perfect innings of cricket, viz., 

 one for each player of the eleven, one for the extras, and 

 one for the " not out" man. This last chance must clearly 

 be considered as distinct from the several chances of the 

 eleven. One man must carry out his bat : we cannot tell 

 who may be the man, but whoever it may be, he clearly will 

 have had two chances to contend against ; first, the chance 

 against his making his double figure before yielding to the 

 enemy's attack, and secondly, the chance that the last of the 

 wickets will fall, though he maintain his own defence, 

 before he has attained double figures. We may fairly 

 regard this extra chance as equal to the average chance of a 

 member of the eleven getting into double figures, since the 

 " not out " man may be any one of the eleven. 



In the case of any given innings then, to be played by 

 eleven men of known skill against an eleven whose bowling 

 and fielding abilities are also known, we have in effect, as in 

 the Yarborough problem, a series of thirteen known chances 

 to consider, each one of which must turn out favourably 

 that the innings may be a " perfect " one. We set down 

 our estimate of the chance in the case of each man that he 

 will pass the nine and so enter among the double-figure men. 

 We take the average of these eleven fractions as the chance 

 that the " not out " will have made his double figui'es before 

 the tenth wicket has fallen. And lastly we get a thirteenth 

 fraction to represent the chance that the extras will run into 

 double figures by considering what are the chances in regard 

 to the innings to be made by the eleven as a whole, and 

 what proportion the extras are likely to bear to the total 

 innings. Thus, as in the Yarborough problem, we have 

 thirteen fractions representing the series of chances, all of 

 which must turn out favourabh-, if the innings is to be 

 " perfect," and, mnltii)lying them all together, we obtain the 

 chance of that event. 



Of course for each match played the calculation of the 

 chance of a pgrfect innings would be different. If players 

 of the strongest defensive force form the eleven, even 

 the bowlers being great batsmen, like Steel (A. G.), 

 Studd (C. T.), Ulyett, Emmett, and the champion himself, 

 the chances of a perfect innings are much less than where 

 an eleven, as usual, divides its strength more definitely 

 between batting and bowling, and includes a man or two 

 selected for skill rather in saving than in making runs, 

 or for such quickness as makes a good wicket-keeper so 

 destructive, or a keen point or short-slij) so effective both 

 in saving runs and catching out players. Then the qualities 

 of the opposing eleven have to be considered. They may be 

 such that even an eleven of the most effective batsmen may 

 be compelled to play a defensive game, insomuch that though 

 that game may prove a winning one, it is not likely to run 

 into high figures. 



All one can do in each c;ise is to consider, according to the 

 be.st evidence available, what chance this player has or that 

 of making not fewer than ten runs against the opposing 

 eleven, and then combining the chances by multiplying, as 

 already explained. Thus one might put down the chance 

 of the first players, A and B, as one in two, and of the 

 remaining nine players in order as one in three, three, four, 

 four, five, five, six, ten, ten»; the chance for the not out, 

 whoever he may turn out to be, might be put at one in 

 twenty, and the chance of a double figure for extras at one 

 in two. Then the chance of a perfect innings would be 

 represented by one in the product of the thirteen numbers 

 — two, two, two, three, three, four, four, five, five, six, ten, 

 ten, and twentv. This product will be the product of the 

 numbers 20, 20, 20, 20, 20, and 108, or 345,600,000, so 

 that the odds would be 345,599,999 to 1 against a perfect 



