Solutions of Mathematical ^lejions. 333 



MATHEMATICAL and PHILOSOPHICAL CORRESPONDENCE. 



Question VII. Anfwered by J. F — : — :— :— i?. 



-L 11 E order in which the firft 51 cards are diRributed being of no importance with re- 

 fpeft to the chnnce for trumps, the folution of this queftion will be fomevvhat fimplifieJ 

 by conceiving the hands of die dealer's two adverfaries to confift of the firlt 26 cards dealt 

 off. The chance that thefe may be all of them of the 39 vyhich are not trumps will evidently 



L 39- "8-37 3^- &c '4 _ 25 • 24 • 23 • 22 . . & c 14 _ iq _ i 



' 51 . 50.49. 48 . . &c 26 51 . 50.49. 48 . . &c 40 ~ 580027 "30527!^, 



fo that the odds againft it 316305264110 i; and yet it is J200300 times more pro- 

 bable that the dealer and his partner fliould have the thirteen trumps between them, than 

 that the dealer himfelf fhould hold them all in his own hand. 



De Moivre, in his " Doctrine of Chances," (2d edit. cor. to prob. xix.) gives a theorem 

 refpeding the chance of drawing any certain number of black and white counters, which 

 is applicable in fuch an infinite variety of cafes, that it is perhaps worth mentioning. It 

 may be thus generalifed : — If a and b be the refpedive numbers of things of two different 

 kinds, heaped promifcuoudy together, the following formula will exprefs tlie value of the 

 chance, that out of c of them, taken at hazard from amongft the whole number « ( = a + b) 

 there be found the exaft number />, and no more, of that kind whofe whole number is a : 



f.r— !.<:— 2..&c.(/ terms)Xi/.a'— i.(/_2...&c. (<7—/ terms) X — ."-^^ •^^^-&c. (/.terms) 



II .n — i .n — 2.7; — 3 . . &c. [a terms). 

 wherein </ is = k — c. A proper fubftitution in this expreffion will alfo give us the fame 

 rcfult as above. 



Question VIII. Anfwered by J. F— .—. — •— r. 



LET a and b be the hyperbolic logarithms of a and b. Then will a".v be = h . 1 . a*" 



and I + a , X + — A-J + .v^ 4- — «♦ &c. = a'""; and, in like manner, 



b- , b3 b" X 



l4-b.Ar+--A- 4- -— - a:3 + x" . . . . Sic. = b ; the fum of which 2 + (a + b).v 



a» 4- b» a^+bJ a^+b" ^ , , 



+ — - — x^ 4- ^ ■ A-= 4- x"* . • ■ &c. = c, by the queftion. Hence, putting d = 



e — 2y and a, $, y, J, e, &c =the above co-efficients of the feveral powers of ;v in the latter 



fcries.wegct.= L^_4.,/.+ i?Lll^^, + Ll^iZZ-4^-Z^^. &,. ^s. 



a, a.^ oi> oiJ ' 



however, this feries docs not, when d is great in refpe£l of a, begin to converge till after a 

 confiderablc number of terms, it fecms better to ufc the common tentative procefs, repeat- 

 edly afTuming two values of .v, and applying the following proportion: — Difference of re- 

 fults ; difference of affumed values : : lead error : corredlion for the' ncareft value. — This 

 method, though an indireil one, has certainly great praftical advant-igcs in a variety of 

 cafes wherein the cxprcffions are fo entangled witli furds or unknown exponents as not to 

 be othcrwifc reducible witliout a great deal cf trouble, 



5 Another 



