I have not read the Aaronson's book, but I'm already confident his use of the word "probability" is more confusing than helpful. Okay, the universe is made up of a wacky "configuration space" of "complex amplitudes" (in lay terms, a many-dimensional sea of complex numbers). And, there is a direct relationship between complex amplitudes and the degree of subjective belief of the physicist doing such and such experiment: take the norm of the amplitude, square it, compare that to the other relevant squared norms, and voilà, you have a probabilistic prediction for the result of your next experiment. (I sweep a lot under the carpet, but it really works like this.)

Something to know however: assuming you made no mistake, then the result of the first experiment tells you nothing about the result of any subsequent trials. Here's an example:

Let us send light through a half-sieved mirror. Have detectors on either side of the mirror to detect the light. Now, you can see that each detector detects half as much light as you originally sent (of course). Things gets interesting when you send photons one by one. The set up of the experiment is such that when you make your amplitude calculations, you find that the squared norm of the amplitudes at each detector is the same. Here, this means that when you send a single photon to the mirror, your should expect to see a single detector go off, but you have no idea which one (subjective belief of 1/2 for either side).

But there's more. Once you made the calculation, you know all you could possibly know about the initial conditions of the experiment (according to current physics). Imagine you sent a photon for the first time, and you see it going through the mirror. Will that change anything to your expectation on the second trial? Not one bit. You will still bet 1/2 chances on either side. So this experiment is the perfect coin toss.

Or is it?

Imagine you send twenty photons, and you see they all made it past the mirror. Initially that's one chance in a million. In that case, you should be strongly tempted to expect the 21th to go through as well. The question is why?

If you are absolutely sure you set up the experiment right, then you shouldn't change your mind, and still bet 1/2 on either side. But are you so sure of your accuracy that you could set up several millions experiments, and get all of them right? Probably not. The more likely explanation is that your so-called "half sieved mirror" is just a piece of glass. Maybe you were tired, and piked up the wrong one. Or maybe some graduate student played a joke on you.

Conversely, an actual coin toss could be perfectly fair: there's a lot going on in a coin toss, and depending on the chosen procedure, you can't expect to have sufficient knowledge of the initial conditions to predict anything but 1/2 for heads, 1/2 for tails, even after seeing the the coin landing tails three times in a row. There is, even then, some kind of "frequency property" in the experiment. It's not because the laws of coin tossing are somehow indeterministic, but because the way the coin is tossed systematically hides relevant information about the exact initial conditions.

(The formal Aumann agreement theorem is a bit stronger than that: two perfects Bayesians cannot have common knowledge of any disagreement. In practice, this means you can't start from different priors, because a prior embody background information, which can be shared. I know that there is controversy about "fully uninformed" priors, but in practice, they don't change posterior probabilities very much.)