Newcomb's Problem is silly. It's only controversial because it's dressed up in wooey vagueness. In the end it's just a simple probability question and I'm surprised it's even taken seriously here. To see why, keep your eyes on the bolded text:

Omega has been correct on each of 100 observed occasions so far - everyone [on each of 100 observed occasions] who took both boxes has found box B empty and received only a thousand dollars; everyone who took only box B has found B containing a million dollars.

What can we anticipate from the bolded part? The only actionable belief we have at this point is that 100 out of 100 times, one-boxing made the one-boxer rich. The details that the boxes were placed by Omega and that Omega is a "superintelligence" add nothing. They merely confuse the matter by slipping in the vague connotation that Omega could be omniscient or something.

In fact, this Omega character is superfluous; the belief that the boxes were placed by Omega doesn't pay rent any differently than the belief that the boxes just appeared at random in 100 locations so far. If we are to anticipate anything different knowing it was Omega's doing, on what grounds? It could only be because we were distracted by vague notions about what Omega might be able to do or predict.

The following seemingly critical detail is just more misdirection and adds nothing either:

And the twist is that Omega has put a million dollars in box B iff Omega has predicted that you will take only box B.

I anticipate nothing differently whether this part is included or not, because nothing concrete is implied about Omega's predictive powers - only "superintelligence from another galaxy," which certainly sounds awe-inspiring but doesn't tell me anything really useful (how hard is predicting my actions, and how super is "super"?).

The only detail that pays any rent is the one above in bold. Eliezer is right that one-boxing wins, but all you need to figure that out is Bayes.

EDIT: Spelling

I have found that the logical approach like this one works much more rarely than it doesn't, simply because it appears that people can manage not to trust reason, or to doubt the validity of the (more or less obvious) inferences involved.

Additionally, belief is so emotional that even people who see all the logic, and truly seem to appreciate that believing in God is completely silly, still can't rid themselves of the belief. It's like someone who knows household spiders are not dangerous in any way and yet are more terrified of them than, say, an elephant.

Perhaps what's needed in addition to this is a separate "How to eschew the idea of god from your brain" guide. It would include practical advice collected from various self-admitted ex-believers. Importantly, I think people who have never believed should avoid contributing to such a guide unless they have reasons to believe that they have an extraordinary amount of insight into a believer's mind.

I have found that the logical approach like this one works much more rarely than it doesn't, simply because it appears that people can manage not to trust reason, or to doubt the validity of the (more or less obvious) inferences involved.

Additionally, belief is so emotional that even people who see all the logic, and truly seem to appreciate that believing in God is completely silly, still can't rid themselves of the belief. It's like someone who knows household spiders are not dangerous in any way and yet are more terrified of them than, say, an elephant.

Perhaps what's needed in addition to this is a separate "How to eschew the idea of god from your brain" guide. It would include practical advice collected from various self-admitted ex-believers. Importantly, I think people who have never believed should avoid contributing to such a guide unless they have reasons to believe that they have an extraordinary amount of insight into a believer's mind.

I can understand 'supernatural' in the context of the Game Of Life: the user ('God' if you like) violates the laws of physics (in this case by anomalously switching some cells on or off).

Can't we give pretty much the same definition of 'supernatural' in the real world? (Relative to the 'true laws of physics' whatever they are.)

Perhaps we could say that a supernatural event is one that "increases the algorithmic information content of the universe", though formulating this precisely would require some care.

It's also often used to mean "not material," or "not obeying the same laws as most stuff we see." So the stuff we see around us is the "natural," and things that don't follow natural law are "supernatural."

I suspect most supernatural beliefs are non-reductable just because they're made up by people and people don't generally think in terms of reductionism. I prefer "magic" form for complicated human-interacting ontologically basic things.

I didn't downvote and it's not an inappropriate topic, but the post is a bit rambling. More thought and editing could allow you to figure out what you're really trying to say, and hence tighten up the post.

So from the negative votes I'm guessing that this is not something you guys find appropriate in "discussion"? It would help me as a newcomer if you also suggested what makes it bad :)

I'm obviously new to this whole thing, but is this a largely undebated, widely accepted view on probabilities? That there are NO situations in which you can't meaningfully state a probability?

Actually, yes, but you're right to be surprised because it's (to my mind at least) an incredible result. Cox's theorem establishes this as a mathematical result from the assumption that you want to reason quantitatively and consistently. Jaynes gives a great explanation of this in chapters 1 and 2 of his book "Probability Theory".

But how valid is this result when we knew nothing of the original distribution?

The short answer is that a probability always reflects your current state of knowledge. If I told you absolutely nothing about the coin or the distribution, then you would be entirely justified in assigning 50% probability to heads (on the basis of symmetry). If I told you the exact distribution over p then you would be justified in assigning a different probability to heads. But in both cases I carried out the same experiment -- it's just that you had different information in the two trials. You are justified in assigning different probabilities because Probability is in the mind. The knowledge you have about the distribution over p is just one more piece of information to roll into your probability.

With this in mind, do we still believe that it's not wrong (or less wrong? :D) to assume a normal distribution, make our calculations and decide how much you'd bet that the mean of the next 100,000 samples is in the range -100..100?

That depends on the probability that the coin flipper chooses a Cauchy distribution. If this were a real experiment then you'd have to take into account unwieldy facts about human psychology, physics of coin flips, and so on. Cox's theorem tells us that in this case there is a unique answer in the form of a probability, but it doesn't guarantee that we have time, resources, or inclination to actually calculate it. If you want to avoid all those kinds of complicated facts then you can start from some reasonable mathematical assumptions such as a normal distribution over p - but if your assumptions are wrong then don't be surprised when your conclusions turn out wrong.

The description of the coin flips having a Binomial(n=?,p) distribution, instead of a Bernoulli(p) distribution, might be a cause of the mis-reading.