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♦ KNOWLEDGE ♦ 



[September 1, 1887. 



A "PERFECT INNINGS" AT CRICKET: 

 A CHANCE PROBLEM. 



URING the cricket season of 188G there 

 occurred what has been called a " perfect 

 innings," * every player on the side scoring 

 double figures, while the extras supplied a 

 twelfth double-figure entry. Not one of 

 the players reached the century, so that 

 when the score of the innings was printed, 

 the numbers opposite the names of the players made a neat 

 double figure column. This has not happened half a dozsn 

 times since cricket has been a game. Moreover, the score 

 as actually formed was unique : for every item was above 

 15 (only one was below 211), and that has never happened 

 before. 



In the Times there appeared, a few days after this remark- 

 able game between the Australians and an English eleven, 

 a letter in which the exceptional nature of the innings was 

 commented on, and a comparison was drawn between the 

 occurrence of such an innings at cricket and the occurrence 

 of what is called a Yai'borough hand (or simply a " Yar- 

 borough") at whist. It happened that in the pages of 

 Knowledge, about half a year before, both these unusual 

 events had been considered, though no comparison had been 

 instituted between them : and the writer of the letter in the 

 Times touched on the circumstance that the kind of innings 

 which had been described in Knowledge as altogether un- 

 usual should have presented itself within so short a time. 

 He expressed also the opinion that it must be more unusual 

 than a Yarborough, a point about which there can be no 

 manner of doubt. 



At first there seems to be no resemblance between the 

 chance of a "perfect innings" at cricket and that of a 

 Y^arborough hand at whist ; but, in reality, they both 

 belong to the Sime class of questions in probabilities. 



A Yarborough hand at whist is one in which there ia no 

 card above a nine — in whist estimation, according to which 

 the ace is very much above the nine. The hand is Killed a 

 Y'^arborough, because early in the century Lord Y^arborough, 

 a sporting peer, was in the habit of offering odds of a 

 thousand to one (generally in guineas) that the deal would 

 not give the person with whom he wagered a hand of this 

 objectionable kind. Observe, we say, " the person with 

 W'hom he wagered." Often the wager is wrongly described, 

 as if Lord Yarborough betted against the deal giving a 

 Y^arboi-ough to any of the four players. Had this been the 

 wager. Lord Y'arborough would have lost money over his 

 venture ; as it was, the story runs that he gained consider- 



* The innings in question is worth quoting. It was played by 

 an Eleven of England at Scarborough, September i, 1886, and is 

 not only remarkable as a " perfect innings," but also as the highest 

 innings ever played against an Australian eleven. It was as 

 follows : — 



England. 



W.G. Grace cJarvis b Giffen 92 



Scotton c Palmer b Truuib'.e 71 



Bates c Bruce b Spofforth 6.S 



Barnes c and b Garrett 4.5 



Barlow b Palmer 16 



Ulvett b Palmer 48 



C. I. Thornton st Jarvis b Palmer 22 



Flowers M'llwraith b Garrett 82 



Briggs c Jarvis b Garrett 21 



E. F. S. Tylecote c Palmer b Giffen SI 



W. E. W. Collins not out S6 



E.'ctras 21 



Total 558 



The play of the Australians in the unfinished match was as 

 follows : — 



Australians. 



G. E. Palmer c Briggs b Barnes 22 



S. P. Jones c Tylecote b Barlow . . 24 



G. Giffen c Thornton b Barnes 18 



G. J. Bonnor retired hurt -IG 



J. W. Trumble b Flowers 24 



A. H. Jarvis b Briggs 18 



W. Bruce c Briggs b Barnes 23 



T. W. Garrett c Ulyett b Briggs . . 1 



J. M'llwraith b Barnes 4 



H. J. H. Scott not out 8 



F. R. Spofforth b Briggs 81 



Extras 12 



Total 231 



Second Ix.xixos.— G. E Palmer st Tylecote b Briggs, "5 ; S. P. Jones not out, 

 108 ; G. Giffen not oat, 7 ; extras, 2 ; total, 192. 



ably, and it is certain that if he only laid the odds often 

 enough he must have gained. 



Now at first nothing can seem more absurd than a com- 

 parison between an innings at cricket and a deal at whist. 

 But if we con.sider the two chances compared above, we 

 shall find that they are much more nearly akin than would 

 at first seem possible. In fact they only differ in kind in 

 this, that the chance of a cricketer making double figures in 

 a given match depends on our estimate of his skill and of 

 the skill of the side against which he contends at the 

 wickets, while the chance of any card dealt to a given 

 player being below a nine is definite. Apart from this, the 

 resemblance between the two chances in their nature is 

 rather i-emarkable. 



Let lis consider them separately, avoiding all that is com- 

 plex or otherwise unsuited to these pages. After all, this 

 need not make our study of the sulyect incomplete ; for all 

 who know enough of the science of probabilities to follow 

 complex or technical statements can supply from their own 

 knowledge what we here omit, while to those who cannot 

 all such statements would be unintelligible. 



Suppose we are watching a deal at whist, all the hands 

 being dealt in the usual way, except the hand of the 

 person with whom the wager has been made that he will 

 not get a Y^arborough. His hand, let us suppose, is dealt 

 fiice upwards, as in single-dummy whi.st. Of the thirteen 

 cards in each suit eight fulfil the conditions of the wager, 

 so that there are 32 cards in the pack any one of which will 

 do for the first card dealt to this up-turned hand. The 

 chance, then, that at the first round the bettor will not get 

 an ace, king, queen, knave, or ten is clearly 32 in 52, 

 which we represent as a fraction. Supposing the firet round 

 favourable, there remain 31 cards any one of which will 

 suit, out of 51 unknown cards. The (act that three other 

 cards have been already dealt, face down, in no sense affects 

 the chance of success at this second round : all we have to 

 consider is that a card has to be dealt which may be any one 

 of the 51 unknown or undisclosed cards. For this round, 

 then, the chance of success is 31 in 51, which again we 

 represent as a fraction. The chance for the third round is 

 30 in 5(1 ; for the fourth, 29 in 49 ; and so on till we come 

 to the thirteenth round, for which — always supposing that 

 all the preceding rounds have been fjivourable for the player's 

 wager — the chance will be 20 in 41), or exactly one-half. 



Now, it is a well-known rule in probabilities that to find 

 the chance of a series of events coming off in the way above 

 imagined, we must multiply together the chances of the 

 several events of the series. Thus, the chance of head being 

 tossed at a single trial being one-half, the chance of tossing 

 head twice in two trials is half of half, or a fourth ; the 

 chance of tossing heads thrice in three trials is half a fourth, 

 or an eighth ; and so on as far as we please, till (for example) 

 we find the chance of tossing head twenty times in twenty 

 trials to be less than one-millionth. Applying this rule to 

 the Yarborough wager, we find the chance of a Y'arborough 

 hand represented by a fraction whose numerator is obtained 

 by multiplying together the thirteen numbers from 20 to 32, 

 both inclusive, while its denominator is obtained by multi- 

 plying together the thirteen numbers from 40 to 52, both 

 inclusive. If the reader cares to carry out this multiplica- 

 tion (fir.st, however, striking out all that is common to both 

 series), and then to divide the large denominator he will get 

 by the smaller (though still considerable) numerator, he will 

 get the fraction 1 divided by 1,828 and something over. 



It follows that at any given deal a player at whist has 

 rather less than one chance in 1,828 of that kind of hand in 

 which every lover of whist delights so greatly — a hand in 

 which there is no card above a nine. The odds against this 

 pleasing result are therefore rather more than 1,827 to 1 ; 



