By trivial argument (of the kind employed in algorithm complexity analysis and cryptography) that you can just toss a coin or do mental equivalent of it, any guaranteed probability nontrivially >.5, even by a ridiculously small margin, is impossible to achieve. Probability against a random human is entirely irrelevant - what Omega must do is probability against the most uncooperative human being nontrivially >.5, as you can choose to be maximally uncooperative if you wish to.
If we force determinism (what is cheating already), disable free will (as in ability to freely choose our answer only at the point we have to), and let Omega see our brain, it basically means that we have to decide before Omega, and have to tell Omega what we decided, what reverses causality, and collapses it into "Choose 1 or 2 boxes. Based on your decision Omega chooses what to put in them".
From the linked Wikipedia article:
More recent work has reformulated the problem as a noncooperative game in which players set the conditional distributions in a Bayes net. It is straight-forward to prove that the two strategies for which boxes to choose make mutually inconsistent assumptions for the underlying Bayes net. Depending on which Bayes net one assumes, one can derive either strategy as optimal. In this there is no paradox, only unclear language that hides the fact that one is making two inconsistent assumptions.
Some argue that Newcomb's Problem is a paradox because it leads logically to self-contradiction. Reverse causation is defined into the problem and therefore logically there can be no free will. However, free will is also defined in the problem; otherwise the chooser is not really making a choice.
That's basically it. It's ill-defined, and any serious formalization collapses it into either "you choose first, so one box", or "Omega chooses first, so two box" trivial problems.
Free will is not imcompatible with Omega predicting your actions(Well, unless Omega delibrately manipulates your actions based on that predictive power, but it's outside the scope of this paradox), and Omega doesn't even need to see inside your head, just observe your behavior until it thinks it can predict your decisions with high accuracy only based on the input that you have received. Omega doesn't need to be 100% accurate anyway, only do better than 99,9%. Determinism is also not required, for the same reason.(Though yeah, you could toss a quantum coin...
One day, you and the presumptuous philosopher are walking along, arguing about the size of the universe, when suddenly Omega jumps out from behind a bush and knocks you both out with a crowbar. While you're unconscious, she builds two hotels, one with a million rooms, and one with just one room. Then she makes a million copies of both of you, sticks them all in rooms, and destroys the originals.
You wake up in a hotel room, in bed with the presumptuous philosopher, with a note on the table from Omega, explaining what she's done.
"Which hotel are we in, I wonder?" you ask.
"The big one, obviously" says the presumptuous philosopher. "Because of anthropic reasoning and all that. Million to one odds."
"Rubbish!" you scream. "Rubbish and poppycock! We're just as likely to be in any hotel omega builds, regardless of the number of observers in that hotel."
"Unless there are no observers, I assume you mean" says the presumptuous philosopher.
"Right, that's a special case where the number of observers in the hotel matters. But except for that it's totally irrelevant!"
"In that case," says the presumptuous philosopher, "I'll make a deal with you. We'll go outside and check, and if we're at the small hotel I'll give you ten bucks. If we're at the big hotel, I'll just smile smugly."
"Hah!" you say. "You just lost an expected five bucks, sucker!"
You run out of the room to find yourself in a huge, ten thousand story attrium, filled with throngs of yourselves and smug looking presumptuous philosophers.