Mathematical_Recreations-Kraitchik-2e

Probabilities 137
with a throw of tails, so A must throw again. In half of these
games we may expect the coin to fall heads, and for each of these
games B will receive $2. Thus again he may expect 500 million
dollars. The remaining 250 million games begin with two throws
of tails. In half of these we may expect heads on the third throw,
and for each of these 125 million games B receives $4, again a total
of $500 million. We may continue in this way for thirty times in
all (since 2 10 = 1024, 10 9 is approximately equal to 2 30 ), so that for
all the games B will receive about $15 billion. Luck may aid him
and increase this amount, or it may abuse him and decrease it;
but if he stakes $15 a game he has a good chance not to lose.
If A has to play only a hundred games the chances are different.
B, staking $15 a game, has more chance to lose. The conditions
of the game are still to his advantage, but his advantage comes
from possible large profits whose probabilities are small. But if A
promises to play 10 12 games instead of 10 9 , B could risk $20 on
each game instead of $15 (10 12 = 240, approximately), with a good
chance of recovering $20.10 12 , without counting in this approximation
the possibility of winning immense sums, which should be
kept in mind in accurate calculations.
The most singular answer to the supposed paradox is that
of Daniel Bernoulli, who was the first to recall the problem
from oblivion. It was originally proposed by his cousin
Nicholas.
According to Daniel Bernoulli, $100 million added to an
already acquired fortune of $100 million is not sufficient to
double the original fortune. What new advantages can it
procure? Accordingly he replaces the notion of mathematical
expectation by that of moral expectation, in the calculation
of which the worth of a fortune depends not only on
the number of dollars it contains, but also on the satisfactions
that it can give.
Here is the solution proposed by Bernoulli. If a given
fortune x is increased by an amount dX, the worth of the increase
is dX.
x
Hence if my fortune increase from an amount