on some Questions in the Theory of Chances. 147 



the individual being alive to receive it, therefore, these tables 

 give the value of unity to be received after any number of years. 

 Considering duration of life to be valuable in proportion to its 

 length, the value of the expectation of life to any individual is 

 the sum of the chances of his living any number of years mul- 

 tiplied by the intervening time, so that if P„ be the chance of 

 an individual living exactly n years, the value of his expectation 

 of life, is 2wP„, which is evidently equal to 2P'„, if P'„ be the 

 chance of an individual surviving n years; therefore, the value 

 of the expectation of life of any individual is the sum of the 

 numbers on the same line in Tables I. and II. The unity of ex- 

 pectation is here the expectation of an individual who is certain 

 to live exactly one year. The Tables I. and II. give the values 

 of contingencies depending on a single life, without discount ; 

 the Tables IH. and IV. are the same values, discounted at the rate 

 of 3 per cent, comjiound interest. These tables give the values 

 of annuities about 6 j)er cent, higher than those calculated from 

 the Northampton, and given by Dr. Price, Vol. II. p. 54. The 

 only tables that I have met with of annuities on female lives 

 are calculated /iroin observations in Sweden, and are given by 

 Dr. Price, Vol. II. p. 422. But they are calculated at 4 and 5 

 ])er cent, interest. It is not to be expected, however, that tables 

 calculated from observations made in one country will serve in 

 another, or even in different parts of the same country. * 



The probability of having «i balls of the first colour in w, + A'^ 

 trials, the colours of the other iV balls being any whatever, is ' 



fx;"'-"- ' (1 - x,]''x.;"-x/'' (1 -Xi-X2....x^_-,)'"pdx,dx,....dXp_i 



fXi'"'X2'"i (1— .r, — J72 Xp_i)'"vdXidxo dx^^^ 



' Since writing the above, I find that Mr. Finlaison has given the values of Annuities, 

 distinguishing the Sexes, in the Report of tlie Comaiittce on Friendly Societies, 1825, p. 14(i. 



T 2 



