I'm pretty certain at this point that most of the confusion results from mixing up probability and subjective anticipation. Subjective anticipation is a loose heuristic, while probability is much more clearly an element of (updateless) normative decision criterion. In particular, observations update anticipation, not probability.

Sometimes, anticipation conflicts with probability, and instead of seeing them as opposed, it's better to accept that anticipation is indeed so and so, and if you were treating it as probability, you would make such and such incorrect decisions, while probability is different, and gives the correct decisions.