DANIEL BERNOULLI. 219 



387. Daniel Bernoulli also inquires what capital the in- 

 surance company must have so that they may safely undertake 

 the insurance. Let y denote the least value of the capital ; then 

 y must be found from 



This is merely the former equation with y in place of a + ic — e. 

 Thus, taking the same example as before, we have^ = ltt2^3. 



888. Daniel Bernoulli now lays down the important principle 



that it is more advantageous for a person to expose his fortune 



to different independent risks than to expose it all to one risk. 



He gives this example : suppose a merchant to start with a 



9 

 capital of 4000, and that he expects 8000 by a ship ; let — 



be the chance of the safe arrival of the ship. The merchant's 

 fortune 'physiqiie is thus 



(4000 + 8000)T^ (4000)^=10751 approximately. 



But suppose him to put half of his merchandize in one ship 



and half in another. The chance that both ships will arrive safely 



81 

 is r7^\ the chance that one of the two will amve safely is 

 100 '' 



9 1 18 



2 X Y^ X — r , that is —— ; the chance that both will be lost is 



1 



r— X . Hence the merchant's fortune 'physique is 



(4000 + 8000)tV(7 (4000 + 4000)^^ (4000)^^= 11033 

 approximately. 



Subtract the original capital 4000, and we find the expectation 

 in the former case to be 6751, and in the latter to be 7033. 



Daniel Bernoulli says that the merchant's expectation con- 

 tinually increases by diminishing the part of the merchandize 

 which is intrusted to a single ship, but can never exceed 7200. 



9 



This number is — of 8000 ; so that it expresses the Mathematical 



expectation. The result which Daniel Bernoulli thus enunciates. 



