150 Mr. Lubbock on the Calculation of Annuities, and 



and if a;^ were the value of ^p at ^j places, x^,^ at 92 places, &c. 

 the law of possibility might be determined approximately by 

 considering (ppX^ as a parabolic curve, of which Xp is the abscissa 

 passing' through the points, of which the ordinates are 



qi + qi + &c. ' 91 + 92 + &C. ■ 



3. In the preceding investigations, the results of the pre- 

 ceding trials are supposed to be known ; it may be worth while 

 to examine what the jirobability of any future event is when 

 the results of the preceding trials are uncertain. 



Let a bag contain any number of balls of two colours, white 

 and black, suppose m trials have taken place, and let e„ be the 

 probability that a white ball was drawn the n'" trial, /„ the 

 Ijrobability that a black ball was drawn. 



e„ +/„ = 1. 



First let e,, e., e„ be all equal, and let x be the probability 



of drawing a white ball. If a white ball was drawn every time 

 in the m trials which have taken place, the probability in n, + n„, 

 future trials of having m white balls, and «o black balls, is 



(n,+n^){n,+n,- l)...{ni + l) fx"'+"' {\—x)"'dx 

 1 . 2 n„ ^ J'x"'dx 



But the probability that a white ball was drawn every time is C" ; 

 therefore, the probability of drawing a white ball n times, and 

 a black ball n^ times on this hypothesis, multiplied by the pro- 

 bability of the hypothesis, is 



(«i+»2) ( w, +w,-l)...(ni + l) ^^ Jor*'''{\~xy^dx 

 1 . 2..' n„ fx"'dx ' 



and the probability of drawing «i white balls and »», black balls 

 will be the sum of the probabilities on every hypothesis, mul- 

 tiplied respectively by the probability of the hypothesis, which is 



