nv THE RISKS OF LOSS OR GUN. 99 



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Let the game be as follows : The events which may 

 happen at every trial are Ej, of which the chance is J ; 

 E ; of which the chance is | ; E 3 , of which the chance 

 is -J ; and so on, ad infinitum. And one of these must 

 occur. The bank engages to give 2/. if E should turn 

 up, 4/. for E 2 , 8/. for E 3 , l6l. for E 4 , and so on, 

 ad infinitum. What should the player give to the bank 

 for one trial ? Write the several possible gains in a row, 

 and underneath each the chance of its being won, as 

 follows : 



2 4 8 16 32 64 128, &c. 

 i i * * & A ik &c. 

 Multiply each gain by the chance of gaining it, and 

 each result is 1 ; consequently the mathematical expect- 

 ation of the player is unity repeated an infinite number 

 of times, or an infinite amount. No sum, then, how- 

 ever great, can compensate the bank for its risk. The 

 Petersburgh problem realises the preceding supposition as 

 follows : A halfpenny is tossed up until a head arrives, 

 which is the event in question. If this happen at the 

 first toss, the player receives 2/. ; if not till the second, 

 4/. ; if not till the third, 8/., and so on. Now, H 

 standing for head and T for tail, the chance of H is ; 

 of TH, |; of TTH, $; of TTTH, ^ and so on. 

 But can it be believed, that if I am only to throw until 

 head arrives, and to receive 2/., or 4/., or 8L, &c. ac- 

 cording as this happens at the first, second, third, &c. 

 throw can it be believed, you will say, that this pros- 

 pect is even worth 100/. ; and is it not altogether mon- 

 strous to say that an infinite amount of money ought 

 to be given for it ? 



Firstly, I will advert to a large number of trials 

 which was actually made. Buffon tried 2048 experi- 

 ments, or sets of tosses, the results of which were as 

 follows: In 106l, H appeared at the first toss; in 

 H 2 



