And with the Omega that designates "the trillionth digit of pi is even" as heads, you want to behave differently.

Specifically, you want to bet on 'heads'. The trillionth digit of pi is a two.

I think we need to find a trickier logical uncertainty as a default example. There is a (mildly) interesting difference between logical uncertainties that we could easily look up or calculate like "Is 1,033 a prime?" or "is the trillionth digit of pi even?" and logical uncertainties that can not be plausibly looked up. Both types of uncertainty are sometimes relevant but often we want a 'logical coin' that isn't easily cheated.