Archive image from page 446 of The cyclopaedia; or, Universal dictionary. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature cyclopaediaoruni07rees Year: 1819 DOCTRINE OF CHANCES. zd and 3d times, may be found by resfbotng ?.s in the fc'.u- n ~\3 I tion of the preceding problem. = ' x - ; the proba- n \ u bi'ity of throwing it thf 7th time, depending on the contin- gency of his having miffed it the id and 41I1, of IVs having miffed it the 2d and 5tli, and of C's having it the ;d n 1 |6 1 and-6th times, may be found = >C -, and fo on. a n In -like maaner t
Archive image from page 446 of The cyclopaedia; or, Universal dictionary. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature cyclopaediaoruni07rees Year: 1819 DOCTRINE OF CHANCES. zd and 3d times, may be found by resfbotng ?.s in the fc'.u- n ~\3 I tion of the preceding problem. = ' x - ; the proba- n \ u bi'ity of throwing it thf 7th time, depending on the contin- gency of his having miffed it the id and 41I1, of IVs having miffed it the 2d and 5tli, and of C's having it the ;d n 1 |6 1 and-6th times, may be found = >C -, and fo on. a n In -like maaner the probability of IV. throwing it the 2d, â l. . ' ' ' i) ' - 3 1 cth, c. times, will be = , 1 , &c. and the pro- J tin n~ I lability of C's throning it the 3d, 6th, Sec. times will be = Hence the whole expedition of A â1, the whole expe&ation of B = â 11 S . n . n â S and the whole expectation of C = â x ' n S . n â ) > + ':âi] +,&c. n > «' â n â 1 I Corollary I. If the folid be a cube, the odds in favour of A agaiuft B will be as 36 to 50, and the odds in his favour againlt C will be as 56 to 25. And fuppoflng the fwm S to be 10I. the values of their refpeftive expectations will be 5I. 1 )S., .jl. 6s., and 2I. 15s. Corollary 2. If, in (lead of a cube, the folid be a counter with two faces, n in this cafe will be =: 2j and the rdds in f wour of A aaaint B will be as 2 to 1, and the odd6 in his favour againft C will he as 4 to I ; that is, their refpective expectations in the fum S (fuppofing it to be icl.) will be el. 14s., 2I. 17s., and 1. 9s. nearly. If, on the contrary, the folid have a great Dumber of faces, the chances will be nearly equal. Hence the advantage of having the priority in the throws will be greater or lef- in proportion as the faces are few or many in number. Corollary 3. If inftead of three there he d perfons to throw the folid fucccilivcly, the expectation of the ill will be S S . »J -; of the 2d =s -j of the 3d = ; of the 4th = nâ
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