::Okay, so unpack "ungrounded" for me. You've used the phrases "probability" and "calculated or measured likelihood of heads coming up", but I'm not sure how you're defining them.::
Ungrounded: That was a good movie. Grounded: That movie made money for the investors. Alternatively: I enjoyed it and recommend it. -- is for most purposes grounded enough.
::I'm going to do two things. First, I'm going to Taboo "probability" and "likelihood" (for myself -- you too, if you want). Second, I'm going to ask you...
"Suppose our information about bias in favour of heads is equivalent to our information about bias in favour of tail. Our pdf for the long-run frequency will be symmetrical about 0.5 and its expectation (which is the probability in any single toss) must also be 0.5. It is quite possible for an expectation to take a value which has zero probability density."
What I said: if all you know is that it's a trick coin, you can lay even odds on heads.
"We can refuse to believe that the long-run frequency will converge to exactly 0.5 while simultaneou...
Maybe I'm stupid here... what difference does it make?
Sure, if we had a coin-flip-predicting robot with quick eyes it might be able to guess right/predict the outcome 90% of the time. And if we were precognitive we could clean up at Vegas.
In terms of non-hypothetical real decisions that confront people, what is the outcome of this line of reasoning? What do you suggest people do differently and in what context? Mark cards?
B/c currently, as far as I can see, you're saying, "The coin won't end up 'heads or tails' -- it'll end up heads, or it'll end u...
GBM:: ..That said, when I say a die has a 1/6 probability of landing on a 3, that means: Over a series of rolls in which no effort is made to systematically control the outcome (e.g. by always starting with 3 facing up before tossing the die), the die will land on a 3 about 1 in 6 times.::
--Well, no: it does mean that, but don't let's get tripped up that a measure of probability requires a series of trials. It has that same probability even for one roll. It's a consequence of the physics of the system, that there are 6 stable distinguishable end-states and explosively many intermediate states, transitioning amongst each other chaotically.
Conrad.