"W. D. Allen Sr." wrote:
>
> It's been years since I was in school so I do not remember if I have the
> following statement correct.
>
> Pascal said that if we know absolutely nothing
> about the probability of occurrence of an event
> then our best estimate for the probability of
> occurrence of that event is one half.
>
> Do I have it correctly?
You may - somebody certainly said it. Laplace is the name that springs
to my mind, and I'm not at all certain about it. But whoever it was was
wrong. In such a case there is no "best estimate", all estimates are
equally silly. The estimate above gives inconsistent probabilities to
the propositions "this urn contains a ball", "this urn contains a red
ball" and "this urn contains a black ball".
One approach to pathological cases like this might be to replace
"probability" by "odds" and to give odds of 0 to 0 for such an event. In
such an approach, odds of (say) 1 to 1 would be vaguer than odds of
23.51 to 23.51, and probability would be the asymptotic limit of odds of
np to nq as n -> infinity.
Probability can be informally defined as the value p such that you can
bet p-epsilon against the opponent's 1-p+epsilon and win in the long
run, no matter how small epsilon may be. Perhaps we could define "odds"
in this sense similarly, with the following changes:
(1) the opponent is omniscient except for the outcome of the plays.
(EG: in most jurisdictions the casino knows (to a high degree of
accuracy) the odds on its slots and the punter does not.)
(2) the opponent gets to choose a side of the proposition
(3) you derive one unit's worth of pleasure from winning (or, more
objectively are allowed a *fixed* bonus of 1 unit if you win in
recognition of (2); or you are the "house" and can add 1 unit to your
opponent's bet as your percentage on the wager. )
Thus, odds of 0-0 would mean that you would know so little about the
situation that "the fix could already be in" either way. In such a
situation you would not take bets on either side of the proposition if
you suspected your opponent was savvy. These are the
correct odds in the situation described above.
Odds of 1-1 would mean that you would accept a long sequence of 2-1
bets on either side; odds of 20-20 that you would accept a long sequence
of 21-20 bets on either side; and so on. In the limit, you obtain
probabilities, when your knowledge of the situation is also absolute and
you can profit from any deviation whatsoever from reciprocal odds.
I have not worked this idea through; if it works, it might be useful,
but it may also involve equally fatal paradoxes if pushed a bit farther.
-Robert Dawson
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