Archive image from page 448 of The cyclopaedia; or, Universal dictionary. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature cyclopaediaoruni07rees Year: 1819 DOCTRINE OF CHANCES. Now fince tlie fmalleft number of points that can + &c be thrown with thofe folids is m, the next m + I, the third m + 2, and (6 on ⢠it follows that the firft term of the pro- duct of thefe two ftries, or unity, will reprefent the number of chances for throwing m point?, the fecond term the num- ber of chances for throwing m -\- i points, the third term the number of chances for throwing
Archive image from page 448 of The cyclopaedia; or, Universal dictionary. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature cyclopaediaoruni07rees Year: 1819 DOCTRINE OF CHANCES. Now fince tlie fmalleft number of points that can + &c be thrown with thofe folids is m, the next m + I, the third m + 2, and (6 on ⢠it follows that the firft term of the pro- duct of thefe two ftries, or unity, will reprefent the number of chances for throwing m point?, the fecond term the num- ber of chances for throwing m -\- i points, the third term the number of chances for throwing m + 2 points, &c. and that the term of the feries, in which the exponent of r is p â m, will reprefent the number of chances for throwing p points. But fince this term arifes from fuch terms of the firft feries. i I the exponents of r being = p p â mânp â m â 2tt, Sec. are refpedtively multiplied into the firft, fecond, third, &c. terms of the fecond ferits, and lince thelall fadtorof the co-efficknt of the term, whofe exponent is p â m, is both in the numerator and denominator âp â m, it is e dent that the whole co-efficient may be divided into â p- - â pâ a+± â ,-2(m) âV ~ l) »â» â3 (m â 1) and the 1. 1. 3. 4. («sâ 1) co-efficients of the terms in which the exponents of r are 2 11, p â m Sec. will be p â in â 1 (m â I) p â 3/1 3 ⢠(''- 0 . i-2-3 0»- 1) &c. Hence, if the firft of the terms jutt mentioned be multiplied into 1 ; the fecond into â m r' ; the third into m . m â I , r and p â n be made â a, p â 2 n = t p â 3 11 =f, Sec. the number of chances for throwing ex- actly p points will be equal to â x ~ ~ J- 1 d-2 d cf points required becomes a mean between the greateft and lea It poffible number that can be thrown on the folids, and that they then as regularly decreafc till the number of points required be the lcalt than can be thrown on thofe folids ; it will be belt, if the number required be nearer
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