Jdne 2, 1882.] 



• KNOWL.EDGE ♦ 



15 



Now let us apply this method to a more f^eneral case. Suppose 

 one bag contains p halls, of which q are white, and that another baf,' 

 "ontains )/ balls, of which q' are white ; a white ball is drawn — what 

 IS the chance that it came from the foi-mer bag ? 



Ucre w<- take ;j ji'. the common multiple of p and p, and replace 

 the lir.st ba;,' by nne containin<^ pp' balls, of which p'q are white ; the 

 second by a pp' balls of which pq' are white. The numbers are now 



equal, and, therefore, our former rule gives ^-i — , as the chance 



pq+p'q 



that the drawn white ball came from the first bag ; and — — ^ 



pq+pq 

 as the chance that this ball came from the second bag. 



If we divide the numerator and denominator of these expressiops 

 by pp' we obtain expressions for these chances which are readily 

 interprctable into a law for all such cases. The former expression 

 becomes 



the latter becomes 



and since - is the probabilitv of drawing a wliite hall from the 



I> 

 first bag if this bag is selected, while 2- is the probability of draw- 



P' 

 ing a white ball from the second bag if selected, we have this 

 general law : — 



If the chance of drawing a white ball from first bag is Cj 

 and the chance of drawing a wliite ball from the second is Cj, 

 then if a white ball is drawn, the chance that it came from the 

 first bag is 1 ;.- , the chance that it came from the second is 



C, 



C, + Co 

 But the bags of balls are merely illustrative, and we 



obviously proceed at once to this general law : — 



If there are two hypotheses equally likely, and one of which miiat 

 be true, and on the first hypothesis the chance of a certain event is 

 t'l, while on the second hypothesis the chance of the event is Co, 

 then, if the event happen, the probability that the first hypothesis 



' and the probability that the second hyj)0- 



is the true 



C, + Co 

 thesis is the true one is . 



C, + Co 



Tlie importance of this formula will be more readily understood 

 when it is applied to illustrative cases, to bo considered in our next 

 number. 



PROBLEMS. 



Problem 43. — A tapering beam is 30 ft. long. At a distance of 

 10ft. from the thick end it is in er|uilibrium. The fulcrum is 

 shifted 2 ft. nearer the small end, and the beam is then in equi- 

 librium when a weight of 601b. has been suspended to the thin end. 

 Find the weight of the beam.— W. D. B. 



Problem 44. — There are two drums ; the diameter of the larger 

 is, say, 3 ft. ; that of the smaller is, say, 2 ft.; the distance from 

 centres is, say, 10 ft. ; required the exact length of belting neces- 

 sary for them.— W. D. B. 



[40]. — The equation belongs to a well-known class. It may bo 

 depressed by putting p for — , when it becomes 



dt 



dr, dO dp 



"""'di-dT-T/ 



d_p_\d(£) 

 "dU~-J, do ' 



substituting and multiplying by 2 wo get 



d{p') 



—i^g- -t-2o/p'= — 2(/0, a linear equation of the first order and first 



degree. 



whence ji' - — 



Therefore l = '!L. 



^^"-L) 



terms.—J. U. C. 



nnot be integrated in a finite scries of 



#ur ©abisft Column. 



Bv " Five of Clubs." 



A GAME FOR STUDY. 

 I HE following hands are given by Clay to illustrate : 



rather, a class of cases — where it is necessary to disregard 



SiNGiLAR Hand a.nd Singclar Ill-Foetcne ; Yarboroighs. — The 

 following is said to be a remarkable hand of cards dealt to tho 

 Duke of Cumberland, as he was playing at Whist at the rooms at 

 Bath, by which he lost a wager of £20,000, not winning one trick. 

 The Duke's hand consisted of King, Knave, nine and seven of 

 trumps (clubs); Ace and King of diamonds; Ace, King, Queen, 

 and Knave of hearts ; and Ace, King, and Qneen of spades. The 

 Duke leads a small trump. Right hand of the Dnke five small 

 trumps, all the other cards hearts and spades. L>ft hand of the 

 Duke Ace, Queen, ten and eight of trumps ; Queen, Knave, ten, 

 nine, eight, seven, six, five, and four of diamonds. This hand, after 

 winning tho first trick, leads a diamond. The Duke's partner's 

 hand all insignificant cards." The above is extracted from the 

 Kaleidoscope of Feb. 4. 1S23. It may interest the readers 

 of your " Whist Column." Was the Duke's lead judicious ? In 

 January last my partner had a Yarborough hand dealt, and many 

 years ago I was at a party where one was dealt. I imagine few 

 people came across two such. — R. G. 



[In " Coolebs on Whist," there is a somewhat similar case, only 

 instead of failing to make a trick, the holder of the strong hand 

 loses five by tricks. Coclebs says, a lead of trumps from such a 

 hand is wrong. But ninety-nine players out of a hundred would 

 lead trumps; and in my opinion the hundredth would lead wrongly. 

 Wo must not judge by the event in such cases. Tlie whist-player 

 can only jilay according to probabilities ; and tho chances arc in 

 favour of the trump-lead turning out well. It is far more likely. 

 f.'r instance, that, it a Heart is led (the Hearts suit being already 

 established, bo it noticed), tho adversary will ruff it, and perhaps 

 establish a cross ruff, than that tho cards would be so singularly dis- 

 tributed in the other hands, as they wore in this case. I suppose, 

 for example, that tho cards had lain thus: — The Duke's hand, as 

 above (call it A'a hand) ; B's, small cards, no tnimps ; Ps hand, the 

 four trumps named above, no spades, four hearts and diamonds ; Z'a 

 hand, five remaining trnmps, no hearts, three spndea and diamonds. 

 Then, if --I leads from his long suit, ho loses two by tricks, which, 

 with such a hand, and a plain suit lead, is singular ill-fortune. 



Yarboroughs are more common than our correspondent supposes. 

 Within the Inst eight months wo have come across three. Tho 

 editor of tho Wcstmimtcr Payers mentions two as occuiring to him- 

 self at the same sitting. — Five of Clubs. 



0UV Cbcsfs Column. 



PRonLEM Xo. 30, p. 5S0 



1. KttoKtG(ch) BPtakosKtor 



2. Q to IW(ch) B to Ktsq 



3. Q takes P(ch) K takes Q 



4. B to BG(nmte) 



1. RP takes Kt 



2. QtoRI(ch) Klo Ktsq 



3. Kt to Q7(nu>to) 



