The doctrine of chances: or, a method of calculating the probability of events in play / By A. De Moivre.

  • Abraham de Moivre
Date:
1718
Catalogue details

Licence: Public Domain Mark

Credit: The doctrine of chances: or, a method of calculating the probability of events in play / By A. De Moivre. Source: Wellcome Collection.

    54/210 page 28
    Therefore having written the following general Series, viz] ~r + P + rri Q.+ R + S &c. wherein P,Q, R, S &c. denote the preceding Terms, take as many Terms of this Series as there are Units in i, (for fince b repre- fents the number of Black Counters, the number of drawings cannot exceed b 4- i ) then take for A the firft, fourth, fe- venth &c. Terms.; for B take the fecond, fifth, eighth &c. Terms; for C the third, fixth &c. and the fums of thofe Terms will be the refpeQive Expe&ations of A, B, C; or be- caufe the Stake is fix’d, thefe fums will be proportional to their refpeftive Probabilities of winning. Now to apply this to the prefent cafe, make » = a = 4, b = 8, and the general Series will become ■S- + 4P+^a+fR + |S+AT + -}.V + fX 4* -- Y: Or multiplying the whole by 49 y, to take away the Fra&ions, the Series will be .165 -f- no + 84 4- 56 + + 2-0 + 10 4 4 + *• Therefore aflign to A i6y + y*>+ 10 = 231; to B no + 35 + 4 = 159 > to C 84 + ao + 1 = ioy, and their Probabilities of winning will be as 231, 159, 105, or as •77> 5b 35- If there be never fo many Gamefters A, B, C, D &c. whe¬ ther they take every one of them one Counter or more; or whether the fame or a different number of Counters; the Probabilities of winning may be determined by the fame ge¬ neral Series. REMARK I. TH E preceding Series may in any particular cafe be fliorten’d $ for if a is — 1, then the Series will be 7T X I + I+I + I + I + I+ I Hence it may be obferved, that if the whole number of Counters be exa£Uy divifible by the number of perfons con¬ cerned in the Play, and that there be but one fingle White Counter in the whole, there will be no advantage or difad- vantage to any one of them from the fituation he is in, in refpeft to the order of drawing. If
    digitised image 1
    digitised image 2
    digitised image 3