i. ON PROBABILITY. 
36-9 1 
hich is ——— = =, the r ; robability. 
whichis mopeize 2 he required probability 
19. If we give to s the different values 0, 1, 2,...s; successively, the sum 
of all the s 4 1 resulting series will of course be the probability that the sum 
of the numbers drawn will not be greater than s. In the last example, if we 
collect the coefficients of all the terms to a'° inclusive, we shall find the pro- 
bability of not throwing more than 10 with three dice. It appears to be 
143464104154 21-28 4 36 = 3=90 108 1 
216 =e 5 
This probability is the foundation of the game of Passe diz. 
20. If, instead of thus giving to s values which are increased by unity at 
each step, we suppose that s may have any value whatever between 0 and 
s, the probability must be estimated according to the rules of continuously 
varying quantities. The result in that case will be found identical with that 
given by our series, if in it we suppose s and nz both infinite. The series 
then becomes 
1 PEN 74 ity SL li=D 73 ips ‘ 
eat) —iG-) FE) —eel 
ano) G; ) + 1.2 ( 
and the sum of all these series will be found by integrating this expression 
between the limits 0 and s, which gives 
1 NX 7: ry Li - 1) } 
A HT dail asd ieatascin Tl a — &e. p, ab 
1%) (z )+ 1:2 G 2) : 
the last or I” term here being the last in which Su ({ = 1) is positive. . 
n 
This taken between the proper limits will give the probability that the 
sum of the numbers drawn is contained between those limits. 
21. Laplace applies this equation, Théorie Anal. des Prob. p. 257, to ) 
finding the probability that the sum of the inclinations of the orbits of the 
planets would be contained within given limits, if all inclinations were simi- 
larly circumstanced. 
There are 10 planets besides the Earth, namely, Mercury, Venus, Mars, 
Pallas, Juno, Ceres, Vesta, Jupiter, Saturn, and Uranus ; therefore, i = 10 
and the sum of their inclinations to that of the Earth at the beginning of 
1801 was 82° 16’ 36”, therefore 
s 82-27683 
rE th = 914187. 
Retaining only the first term of the preceding expression, because aw 
8 ; evil 
[= 1 is negative, the probability that the sum of the inélinations of the the 
tup 
orbits of the planets would be comprised between the limits zero and . 
82°-27683, if all inclinations were similarly circumstanced, is ) 
1 
Tom. (01 0p + 235. 
{35.18 (:914187) 00000011235 
The question we have just solved is nearly the same as to determine the 
probability of the losses of an insurance company upon i policies upon 
persons of the same age being contained within certain limits. 
i A