82 



♦ KNOWLEDGE ♦ 



[February 1, 1887. 



■which at least one singleton shall appear in the hand, we 

 form the followins; little sum in addition : — 



Arrangement. 



5; 4; 3; 1. 



6; 4; 2; 1. 



6; 3; 3; 1. 



5; 6; 2; 1. 



4; 4; 4; 1. 



7; 3; 2; 1. 



6; 5; 1; 1. 



7; 4; 1; ]. 



8; 2; 2: 1. 



8; 3; 1; 1. 



7; 5; 1; 0. 



6 ; 6 ; 1 ; 0. 



8 ; 4 ; 1 ; 0. 



9; 2; 1; 1. 



9; 3; 1; 0. 

 10 ; 2 ; 1 ; 0. 

 10; 1; 1 ; 1. 

 11; 1 ; 1 ; 0. 

 12; 1; 0; 0. 



Total 



Grand Total 



One-foiirtli of the 

 number of hands in which 

 arrangement can 



appear. 



20,.527,933,140 



7,464,702,960 



5,474,115,504 



5,038,674,498 



4,751,836,375 



2,985,881,184 



1,119,705,444 



622,058,580 



.305,374,212 



186,617,574 



172,262,376 



114,841,584 



71,775,990 



28,275,390 



15,950,220 



1,740,024 



628,342 



39,546 



.507 



4S,882,413,450 

 Multiply by 4 



19.i,529,663,SU(l 



T— ^ , itc, 



That is to say, there .are 195,529,653,800 po.ssible hands 

 at whist, in which one .singleton at lea.st appears. 



Now the total number of hands possible is no less than 

 635,01.3,559,600. Thus the chance of a hand showing one 

 singleton at least is represented by a fraction of which this 

 number is the denominator, and 195,529,653,800 the 

 numerator, and such a fraction is not much less than a 

 third. Reducing the fraction by division (by 200) we see 

 that it may be very nearly represented by 



977_65 

 317507' 



and on applying to this the method of continued fractious 

 we get 



1_ 1_ j_ 



3+ 4+ 26 + 



showing that the fraction is very nearly repi-esented by j^j. 



Thus each player may expect to have a singleton in his 

 hand four times in every thuteen deals. And he may 

 expect to find a singleton in a plain suit three times in 

 every thii'teen deals. 



Now obviously we must not multiply -^^- by 4 to get the 

 chance that in one band of the four a singleton will appear, 

 for that would give jf , or more than certainty, for such a 

 result, which is absurd on the face of it. The proper way 

 to calculate the chance for four hands is ,as follows : — 



The chance that we do not find a singleton in the first 

 hand we examine is -j^^, and the same with the second hand, 

 the third, and the fourth ; hence the chance that we do not 

 find a singleton in any one of the four is represented (ap- 

 proximately) by 



13 13 13 13 28561 



(the last three digits of numerator and denominator being 

 alike is an odd coincidence). Hence the chance that there 

 will be a singleton in some hand is equal to 



22000 

 28561' 



and the odds in favour of a singleton (one at least) appear- 

 ing, are no less than 22,000 to 6,561, or about 7 to 2. These 



are not the correct odds, however, for four hands resulting 

 from a single deal, because such hands are not independent 

 of each other. If one hand of four dealt has the suits 

 unequally divided, the chances are that there will be con- 

 siderable irregularity in the others. The full treatment of 

 the problem would require more work than the matter is 

 worth. But from an approximative method I find the odds 

 about 5 to 2 in favour of a singleton appearing in one hand 

 at least of the four resulting from a deal. Hence, while the 

 " nine times out of ten " mentioned in Bohn's " Handbook 

 of Games " must be regarded as absurd, the wager oflered 

 by G. B.'s friend was short of the just odds, which are much 

 in favour of a singleton showing somewhere. Yet no one 

 who had not either calculated the odds for a singleton in a 

 given hand (and thence inferred the probability of one singleton 

 at least in four hands dealt) or observed the actual occurrence 

 of singleton hands in a long series of deals, would think they 

 occur so often. I suspect the cause may partly be that, 

 while the player is apt to rejoice at the occ\irrence of a 

 singleton in a plain suit (not, of course, that he would have 

 any idea of such an inicjuity as leading it), he is sure to be 

 disgusted in much greater degree when he finds but a single 

 trump in his hand. His pleasure in the first case, which 

 occurs thrice as often as the other, being neutralised by his 

 triple disgust at a singleton in tramps, the general effect 

 is to greatly diminish the impression which a singleton, 

 regarded per se, .and inde[)endently of its being plain or a 

 trump, would otherwise produce. In like manner the 

 annoyance arising from the recognition of a singleton in an 

 opponent's hand is neutralised by the lively satisfaction 

 arising from the discovery that he has only one truDjp. 

 The explanation may be far-fetched, but the fact to be 

 explained is curious ; it is certain that in about five deals 

 out of seven, on the average, a singleton appears in one hand 

 at least out of the four : it is equally certain that not one 

 whist-player in a thousand would believe this till he had 

 tested the matter statistically. 



CONCISE EXPRESSION IN SCIENCE. 



By W. Cave Thomas. 



S it not time that Scientists should express them- 

 selves more concisely than they are wont on 

 the theory of light ? To speak either of the 

 '• velocity of light" or "of light waves which 

 have travelLd across the illimitable depths of 

 interstellar space," would have been perfectly 

 correct under the Newtonian doctrine, but is 

 incorrect, inapplicable, and misleading when applied to the 

 " undulatory." In the first place, the theory of \ibratory 

 action in an ethereal medium teaches us that there is no 

 matter oj Uylit to travel, and in the second that waves 

 themselves do not travel across space, either limited or 

 illimitable. The vibratory action initiated either by the 

 sun or by the stars is communicated, telegraphed, through 

 space, just as a wave excited at one end of a stretched cord 

 is communicated throughout its length to the other; never- 

 theless, the initial wave does not travel the length of the 

 cord. The inexact language I am alluding to in past and 

 piesent dissertations upon the theory of light makes u 

 thorough muddle of the Newtonian and undulatory hypo- 

 theses. Moreover, there is very much involved in the 

 correct and clear apprehension of the facts we have cited. 

 For if the vibrations that act upon the optic sense, and cause 

 all the various sensations of light, are also those that act 

 upon a sensitised plate, then we can no longer entertain the 

 notion of a something plus a vibration, termed actinic 



