The problem is that we don’t know what P(T | awake) is (the P(A|C) factor in your notation). That’s the whole question at hand. You are implicitly assuming some value for it in this step.
I am applying a theory of probability that I’m taking as reasonable until I reach a contradiction that shows that it isn’t (proof by contradiction). If P is reasonable, it should give a value to statements like P(T | awake), and that value should follow the normal Bayes rules.
I don’t think I ever injected a specific value that wasn’t already there in the assumptions?
No, that’s just Bayes, conditional on another factor, awake (I should have written it last for higher clarity).
Bayes is: P(A|B)=P(B|A)*P(A)/P(B)
Bayes with an extra condition is: P(A|B,C)=P(B|A,C)*P(A|C)/P(B|C)
(or you can define Q(X)=P(X|C) and do standard Bayes with Q(X))
The problem is that we don’t know what P(T | awake) is (the P(A|C) factor in your notation). That’s the whole question at hand. You are implicitly assuming some value for it in this step.
I am applying a theory of probability that I’m taking as reasonable until I reach a contradiction that shows that it isn’t (proof by contradiction). If P is reasonable, it should give a value to statements like P(T | awake), and that value should follow the normal Bayes rules.
I don’t think I ever injected a specific value that wasn’t already there in the assumptions?