Dec. 23, 1881.] 



KNOWLEDGE 



171 



!«) Taking the proper advantage of White's weak move, Black 

 : atens to win a piece bv B takes Kt., or, if the Knight retires, fcy 

 K: toB.6. 



('') This ends the struggle. Black now wins the Knight, for if 

 20. Q. Kt. to B.3 , then 20. Q. takes K., and the Wliite Knight 

 cannot take the Qneen on pain of mate in 4 moves. If 20 . K.Kt. toQ. t 



then P. to K.4. wins. 



Our problem Xo. -t (wronglv numbered 3) in No. 5 is solved by 

 R. to Q.R.8. 



We have received correct solntions from Gamma, Arkansas, 

 E. F. K., Caissa, A Yonng Player, Try Try Again, S. D. P., R. M., 

 Afternoon, Worcester, Etoniensis, D. Sec. Others incorrect. 



Edward Sarpant points ont that Mr. Hcaley's problem is unsound. 

 If black Pawn becomes a Book, white cannot ■win. This is so. 

 The point appears to have been noted in " Westminster Papers " 

 several months later. By putting the black Book's Pa^vi\ one 

 square further forward the flaw is corrected. 



From the Glaiffoic Xcttf, 

 By Mr. C. B. Baxter, Dundee. 

 Black. 



Wliite to plav and mate in three move?. 



r!;is is the problem to which we referred in Xo. 6. 5Ir. Baxter 

 y ns that he had never seen our older problem. We had not 

 posed he had. Xext week, or later, we shall give an instance 

 vliich we were anticipated. He notes the resemblance of Carl 



_-'?rt"3 problem, in Illiistrated London yens for November 26, to 



■ own, which appeared the same day. 



&m ©afti'sft Column. 



By "Five of Clubs.' 



HP. H. points ont a mistake i i our discussion of this matter. 

 • Lord Y. should have wagered 1,827 to 1, not 1828 to 1, the 

 chance being l-1828th, and the odd-s, therefore, 1,827 to 1. Of 

 course " H. P. H." is right. The numbers representing the odds 

 and the chance arc so nearly the same in such a case as this, that 

 WQ were not careful about a point which in others of our papers on 

 chance we have insisted on clearly and often. In dealing with 

 another point, " H. P. H." misconstrues us. He says, " the 

 chance that a Yarborough will not happen in any deal is 

 nof the same as the chance that it will not happen in a 

 given hand in four successive deals ; for in the former case one 

 hand depends on the other to a certain degree, whereas in the latter, 

 the chance of any combination happening is quite independent of any 

 combination which may have preceded it. 1 agree that the chance 



of a Yarborough is ro55i ^^^ consequently the chance of a Yar- 



1 . . 



boTOngh in four consecutive deals is Tcocc Following your prin- 

 ciple, this would be the chance of four Yarboroughs in one deal, 

 which is a manifest absurdity, for we ascribe " (thus) " a mathe- 

 matical chance to a clear impossibility." The question we were 

 really considering was what odds should be offered to each member 

 of a party of four at whist that his hand would not be a Yar- 

 borough; and we (practically) aflirmed that £1,827 to £1 should be 

 offered to each. " H. P. H." seems to consider that this is the 

 same as offering £457 to £1 (roughly) against the occurrence 

 of a Tarborongh in a single deal. But this is not the case. Take 

 a simple case illustrating at once his difficulty and our position : — 



Suppose there are four cards marked respectively A, B, C, D, to be 

 dealt, one to each of four persons. Then the chance that any par- 

 ticular card, as A, will be dealt to any given person of the party of 

 four, is obviously one-fourth, or the odds 3 to 1 against that event, 

 so that with that person any one might at once safely and honestl\ 

 wager £3 to £1 against his getting that card. Xow our position is 

 that the same odds might be offered with each one of the four, 

 although it is certain, in this case, that some one of the four must 

 have card A. (In the Yarborough case it is not certain but more 

 probable that one of the four will have a Yarborough than that 

 any particular one will have such a hand.) Well, " H. P. H." might 

 reason that this is not the case, for if the chance is i that a 

 particular person will have card A in any given deal it is (i)* that he 

 will have it in four successive deals, and on our principle the same 

 is the chance that each one of the four persons will have the card A 

 in a single deal, or, in other words, the odds are only 255 to 1 against 

 the manifest impossibility that each member of the party of four 

 shall have the same card dealt to him out of four. Yet it is per- 

 fectly clear that the just odds are 3 to 1 with each person of the 

 fotir, and the proof is that if these odds are wagered with each 

 the event can bring neither gain nor loss to the laj'er of the odds ; 

 he will have to pay £3 to one of the fom", and receive £1 from each 

 of the others. So, if a person wagered £1,827 to £1 with each 

 of four persons, before an ordinary whist deal, that that person 

 would not get a Yarborough, he would be laying the just 

 odds. Xow let us see what his wager really amounts to in this 

 case. If he loses to one, he loses £1,827. One of the others might 

 have a Yarborongh, but the chance that this would happen is verj' 

 small : it is really this, that out of thirty-nine cards dealt to three 

 persons, one would only receive cards belonging to a particular 

 group of nineteen — a chance very small indeed. Begarding it for 

 the moment as zero, we may say that it is certain, or all but certain, 

 that from each of the remaining three players the layer of the odds 

 will receive £1. Therefore, the layer of the odds pays £1,827 ami 

 receives £3, or loses only £1,824. His case is, therefore, similar to 

 that of one who had laid against a Yarborough occurring in each of 

 four successive deals to one only of the four players, except that 

 this one might have had to pay £1,827 for each of the four deals, 

 whereas the other could only have to pay for two at the outside, an<l 

 would most probably have had to pay for one only. The difference 

 exactly makes up for the interdependence of the four hands in any 

 given deal. 



Take a simpler illustrative case to show what we mean. 

 A person, P, wagers with another, X, one of four to whom four 

 cards, A, B, C, D, are dealt four times running, that X will 

 not receive a particular card A, offering £3 to £1 at each of 

 four deals. Unquestionably each wager isr fair. X may have 

 A each time, in which case P will have to pay £12, or X may 

 not draw A at all, in which case P will receive £t. There are 

 other eventualities easily followed. But the wager is manifestly 

 fair. Xow take a single deal. P wagers with W, X, Y, Z severally 

 £3 to £1 that they will not have card A. In this case, one of the 

 fotir must have the card, and to him P must pay £3, receiWng from 

 each of the others £1, or neither losing nor gaining. ,Since each 

 wager, or rather each set of wagers, is manifestly fair, we see that 

 the possibility in such cases of ha^Tng to pay the odds more than 

 once when successive deals are considered, exactly counterbalances 

 the certainty of winning in some cases (most probably in three 

 when a Yarborough is in question, and certainly in three where four 

 cards only are in question), in the case of wagers with the four 

 parties to a single deal. We have, in fact, only to ask whether a 

 certain wager with one party to the deal is fair or not. If it is 

 fair, we may be well assured that there is no unfairness (either way) 

 if the same wager is made with each of the four players. 



However, although this, and this only, was what we were con- 

 sidering (as should be obvious from our remark beginning, " Sup- 

 posing Lord Y. offered £1,000 to £1 to each member of a whist 

 party for ten deals," &c.), " H. P. H." very naturally misunderstood 

 us, seeing that we wrote,*carelessly, as if we were considering " the 

 chance that a Yarborough will not occur in any given hand " (these 

 are our verv' words, and naturally misled him). The chance of this 

 is not to be inferred so simply, as our words might have suggested, 

 from the true odds against the occurrence of a Yarborough in a 

 single hand. To return to our simpler case. The odds against a card 

 A being dra^vn by one of the four is i, and the true chance of its 

 being dealt once to a given person in four successive deals is (|)* or 



-—r ; so that the odds in favour of its being dealt to him once in 

 2o6 



four deals are 175 to 81. Thus only £81 can be safely wagered 

 against £175 that the card will not be dealt once, at least, in four 

 trials to one of the four players ; but the chance of the card being 

 dealt to one of fonr persons in a single deal is, of course, certainty, 

 or 1 ; so that no sum, however small, can be wagered against any 

 sum, however large, that the card will not be so dealt. 



