If this was true, then we could not observe the event „any other coin sequence“ as well since this event is by definition not being tracked.
When you are tracking event A you are automatically tracking its complement.
In fact, in order to detect a correspondence between a coin sequence that we have in mind and the actual sequence, our brain has to compare them to decide if there is a match. I can hardly imagine how this comparison could work without observing the specific actual sequence in the first place. That we classify and perceive a specific sequence as „any other sequence“ can be the result of the comparison, but is not its starting point.
Oh sure, you are of course completely correct here. But this doesn’t contradict what I’m saying.
The thing is, we observe a particular outcome and then we see which event(s) it corresponds to. Let’s take an example: a series of 3 coin tosses.
So, in the beginning you have sample space which consist of all the elementary outcomes:
And an event space, some sigma-algebra of the sample space, which depends on your precommitments. Normally, it would look something like this:
Because you are intuitively paying attention to whether there all Heads/Tails in a row. So your event space groups individual outcomes in this particular way, separating the event you are tracking and it’s complement.
When a particular combination, say is realized in a iteration of the experiment, your mind works like this:
Outcome is realized
Therefore every event from the event space which includes is realized.
Events and are realized.
This isn’t a rare event and so you are not particularly surprised
So, as you see, you do indeed observe an actual sequence, it’s just that observing this sequence isn’t necessary an event in itself.
And what does it not capture in thirder position, in your opinion?
So thirder think. But they are mistaken, as I show in the previous posts.
Thirder credence fits the No-Coin-Toss problem where Monday and Tuesday don’t happen during the same iteration of the experiment and on awakening the person indeed learns that “they are awaken Today”, which can be formally expressed as an event Monday xor Tuesday.
Not so with Sleeping Beauty, where the participant completely aware that Monday awakening on Tails is followed by Tuesday awakening, therefore, event Monday xor Tuesday doesn’t happen in 50% cases, so instead of learning that the Beauty is awakened today she can only learn that she is awakened at least once.
In Sleeping Beauty problem being awakened Today isn’t a thing you can express via probability space. It’s something that can happen twice in the same iteration of the experiment, just like getting a ball in the example from the post. And so we need a new mathematical model to formally talk about this sort of things, therefore weighted probability space.
I suppose you’ve read all my posts on the topic. What is the crux of our disagreement here?