34 ON PROBABILITY. 
and dividing the quantity so obtained by the integral a 
J wx 2%, ord ds.... Ar - 
taken between the same limits. 
If (m1) (m+ 2) (my, +3)... (my +n) be represented by [m, + 11" 
(me + 1) (my~+ 2) (ma—+3) ... (my + ny) 0 ” [m, + 177 
Cm +)Em-+itl)..(E0m) +2) +i) by [Em) +>" 
= being used as a sign of collection to denote that the sum is to be taken 
of all quantities which are represented by a general symbol, these integra- 
tions give for the probability required, 
Clm, + 171 [meg + 172 .... [m; 4-17" ; 
[mms eovuvnnnens +m] Trem? 
which is the same, with the difference of notation, as if the simple probability 
m, + 1 fi 
f drawi ball of the 7 col — The probabilit . 
of drawing a ball of the 7 colour was REO p y tn 
om 
of drawing a ball of the 7% colour in one succeeding trial is ’ " 
m, + 1 whic 
m, ~-m, Fo Lr +m; + 7 lities 
53. One of the most interesting and useful applications of the theory of Hy 
probabilities is the solution of questions connected with the duration of life Expe 
and the calculation of the values of annuities and reversionary payments. Valu 
The value of an annuity is the value of the sum of the annual payments made or 
to an individual throughout his life. oe 
1 : . 
Let 1 -} rate of interest = = and let p,, , be the probability of a given 
individual aged sm years living at least n years ; the value of any sums to be - 
paid to him at the expiration of n years, neglecting discount, is the value of 
this sum multiplied by the probability of the individual being alive to receive 
it, which is equal to sp, , this must be discounted in order to obtain its 
present value, which reduces it to s 7" p,.,.. If @, is the -value of an oo 
annuity of £1. to be received by an individual aged m years, the value of an 
annuity of £5 to the same individual is s @,, = s = 1" p,, 
54. When an individual insures his life at any office, the insurance com- a . 
pany agrees to pay his executors a certain sum at his death, whenever that or 
event may take place. 
Let ¢u,15 Gm, 90 Guya+ + + + «qm,» De the probabilities that an individual, aged 
m years, dies in the first, second, third, &c., or n® year, then the value of Thus 
£1. to be paid whenever he dies, discounted, is 
Pm FP mato eeinne + Gal a 
but Qm,1 = 1-— Pm1s Gm 2 — Pm, — Pm, 2s pavm 
therefore the value of the insurance is 
7(1 = Pu) + 1° (P,1 = Ps) + &e. 
=r(l + a,) — a,, Va 
and the value of £s to be paid when the individual dies is TV 
sr (1 + a,) —sa,. 
By means of this expression the value of an insurance at the age m may 
be deduced from the value of the annuity, and vice versd. A person insur- 
ing his life, instead of making one payment to the office, generally pays an on