ON PROBABILITY, 15 
22. A shilling is tossed into the air; B gives A a certain sum, in considera- 
tion of which A engages to pay B 2 pounds if the shilling falls head the first 
dhe gy time; 4 pounds if it falls head the second time, and not before ; 2" pounds 
pA if it falls head the n™ time, and not before. 
The expectation of B is 
1 1 | 1 
5b 2X TAXT HBX. +2 X o=Ln 
This is called the Petersburg problem, probably from the mention made 
of it by Daniel Bernoulli in the Transactions of the Petersburg Academy ; 
it was first proposed by Montmort in the Analyse des Jeux de Hasard, and 
WY Ui has been generally considered as involving a great paradox, because if it is 
Mob ror agreed that the game shall not be discontinued till the shilling falls head, n 
" must be made infinite, and the expectation of B is infinite ; still no man of pru- 
¥ dence would be disposed to venture even a small part of his fortune at this 
game. On account of the celebrity of this problem we have inserted it, but 
there is nothing paradoxical in the result more than in any other case of long 
odds : it shows, however, that in order to estimate the value of a contingent 
advantage toc an individual, other elements must be considered, and that the 
moral expectation, as it is called, to distinguish it from the mathematical 
expectation, depends on a great many circumstances which it is very difficult 
to submit to calculation. 
23. It is evident that the value of any sum is much greater to an indi- 
vidual who lives by his daily labour and is without any capital, than it is to 
an individual possessed of £100,000 ; we may, therefore, meet this difficulty 
by supposing that the value of a given sum to any individual, is proportional 
to and may be represented by that sum, divided by the whole of the fortune 
which he possesses. 
Let @ be the sum A is to pay B on the event of which the probability is p, 
wus Dr fier fiatsine Bcsons of oii te wun nngems on sinn re in sinspiseith 
and let f" be the fortune possessed by A. On the preceding hypothesis the 
pp aLH expectation of A is 
thts of bq ap 
(ft+0) kf = BD. 
oe If b g = ap, which must be the case in order that the wager may be fair, 
loge ++ the expectation of A will be > = <0 as 
: >=<0 
f+ F-a 
f-a>=<f+borat+b<=>0; 
2 and b are both necessarily positive, therefore @ - 5 > 0. It is, therefore, 
i evident that according to this theory of the value of money, a wager under 
the most favourable circumstances consistent with honesty, injures the for- 
tune of the gamester, because he loses more by losing than he can gain by 
; gaining any wager, since the amount of it must be compared in the former 
case with his diminished, in the latter, with his augmented fortune. 
24. Suppose the fortunes of A and B to be both equal to f; the expectation 
bq ap 
of A is Te 
ap bg 
of B is UE3" 0-5