144 Mr. Lubbock on the Calculation of Annuities, and 



the Government of this country granted Annuities, the price^ of 

 which depended on the price of Stock, which renders their tables 

 complicated. I have given their values of a deferred annuity 

 for five years, compared with those I have calculated from the 

 observations at Chester; it will be seen that the former are 

 much too high. 



2. Suppose a bag to contain a number of balls of p different 

 colours, and that having drawn 



viy + m^ + m^ + nij, 



balls, wj, have been of the first colour, m., of the second colour, 

 m, of the third colour, m^ of the p'" colour. If Xi, x^, x^ . . . . x, 

 are the simple probabilities of drawing in one trial, a ball of 

 any given colour, the probability of the observed event, is 



«^ 1 1^ **'2 • • • • • *^n P 



multiplied by the coefficient of Xi'^'x^"'" Xp'% in the develop- 

 ment of 



(x, + x^....x;)'"'-*-'"' ■'•■+%. 



The event being observed, the probability of this system of pro- 

 babilities is 



divided by the sum of all possible values of this quantity. 



The probability in n, + Wj + Wp subsequent trials of having 



n, balls of the first colour, n^ of the second, n, of the />"*, is a fraction 

 of which the numerator is the sum of all the values of 



x,"'-^"' X xi"^-*-"^ Xp%+\, 



and of which the denominator is the sum of all the values of 



Xi"*" X x/'' Xp'", ; 



