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Notes -
So, I wanted to give you an opportunity to suggest your own name and conceptual meaning for the number that Alice puts into the computer in Variant 2 before I gave my own take on it. I think at this point, you've had an opportunity and have not taken it, so here goes.
I cannot think of anything that is appropriate to call this number (the number that Alice puts into the computer in Variant 2) other than some form of "the probability that Bob observes an outcome". Alice has to be reasoning about Bob's observation function when she computes this number. It's pretty obvious, because if we keep everything else the same, but fiddle with Bob's observation function, then we see a corresponding change in Alice's computation and the resulting value of this number (and no other change in Alice's reasoning or behavior, as evidenced by her own bets).
As such, if I can put it in a pointed way, why would one think that Alice is smart enough and capable of distinguishing between "the probability that Bob observes an outcome" and "the probability of the coin flip, itself"... but is too stupid to distinguish between "the probability that I, Alice, observe an outcome" and "the probability of the coin flip, itself"?
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