Another Bent Coin Problem

[bigpooch]

Suppose I have a coin and bend it so that the probability
of it landing heads, call that P, if I flip is NOT 1/2
(after all, there might be a reason to bend the coin).

What then?

Without any extra information, the betting line is at 1-1,
yet we don't know on which side we would have the best of
it.

On the one hand, there is some information, but because it
is so little ({1/2} is not only a set of measure zero on
[0,1], but it's a mere "point"), the Bayesians will still
say their value of P is 1/2. The frequentists will say that
we don't know what P is, but it isn't 1/2.

Is my "interpretation" of the common points of view correct?