Yet another question on the meaning of probability... Unfortunately it is ill-posed and has insufficient information for a useful answer.
Part of the problem is the lack of a distinction between probability and randomness.
"In common usage, randomness is the apparent or actual lack of
definite pattern or predictability in information." (Wikipedia)
In other words, randomness is a quality of the process that generated the observations.
Probability is the branch of mathematics concerning events and
numerical descriptions of how likely they are to occur. (Wikipedia)
In other words, probability is a way of modelling random processes.
Now arguably there is no evidence that any process in the real world is demonstrably actually random (note Bell's theorem does not establish that the quantum world is truly random - but does narrow down the options). In that case, all probabilities are epistemic in the sense that they are used to model the effects of deterministic behaviour that are only unpredictable because we lack full knowledge (e.g. if we knew all of the dimensions and forces applied etc. we could predict the result of a dice roll with complete accuracy).
However, when we make a probabilistic model we often talk of "aleatory" uncertainty/probability to represent things that occur "by chance". However what we are really saying is that alleatory uncertainty relates to uncertainty due to factors we are not interested in modelling explicitly and we tend to use "epistemic" probability to discuss uncertainty in factors we actually are interested in, caused by a lack of information. However, ultimately it is all "epistemic" as it is all being used to model things that are unpredictable due to a lack of knowledge.
So let's fisk the question:
The dice then lands on 700,000. Let’s assume we live in a world where
this dice roll is not deterministic. In other words, let’s treat this
as some quantum sort of event that is truly and inherently not
determined with each outcome being as equally probable.
Right, so in this hypothetical there is genuine randomness.
Now, if it lands on that number, clearly this would be considered a
remarkable feat of luck. One might even put a number on the chances of
this: 1 in a million.
Shifting from "probability" to "luck" introduces unnecessary ambiguity, bug or a feature? - you decide. But to stick to probability, it would be reasonable to say that the probability of this outcome is one in a million.
Better still, we could say that U represents our knowledge about the nature of the universe, we could write this more exactly as
P(X = 700,000 | U = random) = 1/1,000,000
In other words the conditional probability of rolling 700,000, given we assume a truly random universe is one in a million. This is a more explicit statement of probability that we would normally use because the condition is "taken as read" and we rarely need to actually say it. However, when faced with an ill-posed question, it can help to really spell it out.
But now let’s suppose the same scenario but assume that we live in a
deterministic world. In other words, the initial conditions and the
laws of the universe just happened to exist in a way where the dice
roll could only have landed on 700,000 and nowhere else along with me
thinking of 700,000 and nothing else.
O.K. so we have a completely deterministic universe, so our knowledge of U is different, so we might say
P(X = 700,000 | U = deterministic) = 1
Note that there is no contradiction here as these are different conditional probabilities, because they represent different assumptions about the nature of the universe.
Technically, the “physical” probability of anything undetermined is 0.
There is no "physical" probability only randomness, but yes
P(X ~= 700,000 | U = deterministic) = 0
Thus, the “physical probability” of anything except 700,000 would be 0
before I even roll the dice (unlike the first scenario).
again, "physical probability" is meaningless - probability is a tool for modelling randomness, whether genuine (if it exists at all) or apparent.
But one can perhaps still give this an “epistemic” probability of 1 in
a million.
What "Hart Lort" has done here is to subtly drop the condition and is describing P(X = 700,000) rather than P(X = 700,000 | U = deterministic), in other words epistemic probabilities are different if they are contingent on different states of knowledge (as you are answering a different question).
Is this scenario more, less, or about as equally “lucky” as the first?
This is unanswerable without additional information. It depends on how likely or unlikely we are to be in a deterministic universe where the dice roll gives 700,000. If there is only one possible universe, then there is no luck involved as there is no randomness. If on the other hand there is a multiverse of deterministic universes where the die gives different values depending on some random process involved in the creation of that universe, then yes, it might be very "lucky".
Let’s make the scenario even more interesting. Let’s assume that we
were in a past eternal universe. In other words, the laws and initial
conditions were simply always set up this way eternally. They never
actually “started”. What then now?
This actually doesn't make a difference, if there is no genuine randomness then there can be no "luck". However the related probabilities are epistemic and depend (are conditional) on your assumptions about the nature of the universe you occupy.
As in all philosophy, there seems to always be a problem with the use of natural language, and they key is to understand the ambiguity and cut through it by explaining what you actually mean (rather than exploiting the ambiguity to make a "debate").
