220 DANIEL BERNOULLI. 



without demonstration is demonstrated by Laplace, Theorie . . . des 

 Froh., pages 435 — 437 ; the proposition is certainly by no means 

 easy, and it is to be wished that Daniel Bernoulli had explained 

 how he obtained it. 



389. Daniel Bernoulli now applies his theory to the problem 

 which is known as the Petershurg Problem, probably from its first 

 appearing here in the Coiiimentarii of the Petersburg Academy. 

 The problem is similar to two which Nicolas Bernoulli proposed to 

 Montmort; see Art. 231. 



A throws a coin in the air ; if head appears at the first throw 

 he is to receive a shilling from B, if head does not appear until the 

 second throw he is to receive 2 shillings, if head does not appear 

 until the third throw he is to receive 4 shillings, and so on : re- 

 quired the expectation of A. 



The expectation is 



1 2 4 8, ... V 



2 + 22 + 2^ + 2^ + • • • ^^^ 'infinitum, 



that is ^ + ^ + 2 + 9 + • • . ^'^^ infinitum. 



Thus ^'s expectation is infinite, so that he ought to give an 

 infinite sum to B to induce B to play with him in the manner 

 proposed. Still no prudent man in the position of A would be 

 willing to pay even a small number of shillings for the advantage 

 to be gained. 



The paradox then is that the mathematical theory is apparently 

 directly opposed to the dictates of common sense. 



390. We will now give Daniel Bernoulli's application of his 

 theory of Moral expectation to the Petersburg Problem. 



Suppose that A starts with the sum a, and is to receive 1 if 

 head appears at the first throw, 2 if head does not appear until the 

 second throw, and so on. ^'s fortune physique is 



{a + 1)^ {a + 2)^ {a + 4)^ (a + 8)^^ ... - a. 



This expression is finite if a be finite. The value of it when 

 a = is easily seen to be 2. Daniel Bernoulli says that it is about 

 8 when a = 10, about 4 J when a = 100, and about 6 when a = 1000. 



