I’m really interested in figuring out your objections to the post, but for now they seem quite incoherent to me. As if some words were wrongly changed by autocomplete, and, in case with this reply, as if your message was sent before you finished writing it.
Could you try once again and present me a simple and and explicit real world example where my math fails, writing your post not from a phone?
I really don’t know why you keep coming up with these alternative formulations of the same problem, an answer to which I’ve already given. Can’t you just apply the logic I’ve very explicitly described in the posts and derive the answer for your modified problem yourself? But surething, here is how one supposed to reason:
There are 4 equiprobable outcomes of the experiment for the Sleeping Beauty:
HH_HT; HT_HH; TH_TT; TT_TH
Meanwhile for the person who randomly observes one of the days of the experiment there are these 4 equiprobable outcomes:
HH; HT; TH; TT
When the person learns that the Beauty is awaken, outcome HH is eliminated and now they are 1⁄3 confident that the first coin is Heads.
See the Observer Problem.
For this problem the conditional probability of HT is indeed 1⁄3
This, however, gives no new information to the Sleeping Beauty herself as she is participating in a different probability experiment—the one with either one or two awakenings, where she observes them in order. She has different sample space and no outcome is eliminated by her awakening.
For this problem HT isn’t an event and, therefore, doesn’t have any probability and the answers to Q3C and Q3D are “No”.