16 CN PROBABILITY. 
In order that the expectations of A and B may be equal 
bq apc Vdp bq 
(fo) (F-oy (Uta (f—D 
2fap_2fbq a_ gq fi-a 
FT RmE 3p Fob 
= >= 2 as fr—at > = < f2— 1? asb > = < a; therefore if @ > b, 
P v 
that is, if p < ¢, (@ and b are the odds) « is not so much greater than b, 
as it would be if the moral expectation were not taken into account. In 
practice we believe this is always the case, as the reader may have observed at 
in the example we took, page 9. 1 
Ex. 14. Let p be the probability of a vessel coming into port, q the pro- 
bability of a total loss, @ the value of that part of the cargo in any vessel, en 
which belongs to A, let there be 7 vessels, and f, as before, A’s fortune. x 
The mathematical value of A's expectation, the n @ goods being equally a 
distributed over zn vessels, is me 
nin —1 car 
nap + n—1Nanp""'g+n=-2)a. 2 Dye ¢ + &e. me 
=nap{p+q}"-'=nap, sinceptqg=1, - 
which is the same as if they were all on board the same vessel ; but, on the ah 
preceding hypothesis, the value of A’s expectation is I 
Pp" n pe 1 q ; n. (n se 1) dip 
aN -Da.—F—— —-2 a. —— 
i Ya none ) 1.2 bet 
" -2 q® Wi 
—_— | &e. nun 
J+r—2)a t taks 
Where every denominator but the first is less than (f + na), and, therefore, i 
this series is greater than the sum of the numerators divided by (f+ n a), the 
a —1 a 
that is, greater than pep. (oto) that is, than lp i 
f+nea f+nra Ise 
The value of A’s expectation is, therefore, greater than oti which is din 
f+nrna pa 
the value of his expectation when all his goods are on board the same vessel. ns 
This principle of the distribution of risk is well known to every merchant, 
and, in fact, affords him all the advantages derived from insurance. It is on on 
this principle that the East India Company never insure their vessels. 
"The hypothesis we have adopted in the preceding problems was suggested 0 
by the celebrated naturalist, Buffon. 
25. Daniel Bernoulli supposes that the value of the fortune of any indi- 
vidual is made up of an infinite number of indefinitely small elements, the 
value of each of which is inversely as the capital already formed, so that if 
¢ represent a small element of the fortune f, and f], f,, &c., its successive 
amounts, the value of the whole fortune f will be 
ko ko ko 
— + — -F......,. + =—— k being some constant quantity. 
When ¢ is indefinitely diminished, this series = % log. (£). The divisor 
1 
f, denotes that part of the fortune of the individual which is absolutely un- 
alienable, and below which his fortune cannot sink. This theory affords us y