A few comments on the way Laplace reported his result in terms
of betting odds are in order.
- First, he does not say that he would be ready to make
a 11,000 to 1 in favor of the result, but rather
that ``the odds are 11,000 to 1''.
This implies that ``11,000 to 1 in favor'' and ``1 to 11,000
against'' are both fair bets. This is essentially the idea behind
the so called de Finetti's coherent bet:
in order to express your degree of belief in favor of something,
you fix the odds and leave somebody else to choose in which
direction to bet. This is the best way to
force people to assess what they really believe,
no matter what the event is and how the probability
has been evaluated (at limit, just by intuitive reasonings,
if no other means are available - why not? what is
important is that once you fix the odds you have no sensitive
preference towards either direction.).
- Second, what is the sense of a bet whose result would have
not probably been solved in Laplace's lifetime? This is
another important ingredient: the fact that bets
have to be considered hypothetical (`virtual').
It is just a way to assess probability.13