Mathematical_Recreations-Kraitchik-2e

136 Mathematical Recreations
makes me very eager that you should give us a solution of
the Petersburg problem, which seems to me insoluble on the
basis of known principles."
Condorcet and Poisson thought that the game contains a
contradiction. A enters into an engagement which he cannot
keep. If heads does not come until the hundredth throw,
then B's profit will be a mass of gold bigger than the sun,
and A has deceived B by his promise.
Bertrand gives the following commentary:
This observation is correct, but it does not clarify anything. If
we play for pennies instead of dollars, for grains of sand instead of
pennies, for molecules of hydrogen instead of grains of sand, the
fear of becoming insolvent may be diminished without limit. This
should not affect the theory, which does not require that the stakes
be paid before every throw of the coin. However much A may
owe, the pen can write it. Let us keep the accounts on paper.
The theory will triumph if the accounts confirm its prescriptions.
Chance will probably, we can even say certainly, end by favoring
B. However much B pays for A's promise, the game is to his
advantage, and if B is persistent it will enrich him without limit.
A, whether he is solvent or not, will owe B a vast sum.
If we had a machine which could throw 100,000 coins a second
and register the results, and if B paid $1,000 for every game, B
would have to pay $100,000,000 every second*; but in spite of
this, after several trillion centuries he will make an enormous profit.
The conditions of the game favor him and the theory is right.
This statement can be explained without extended calculation.
Suppose, to avoid too large numbers, that A agrees to play 109
games. Let us see how much B may expect to receive without assuming
particularly good luck.
"Among the 109 games we can expect that 500 million will end
on the first throw and yield B only $1 apiece. It is not unusual for
heads to fall once in two tries. If the number is less, B is in luck,
which we do not assume. The other 500 million games all begin
* Since B pays by the game and not by the throw, he will not ordinarily
pay the full amount every time; for many of the throws will represent
continuations of uncompleted games.