(Okay, I'll unpack the implication:) Assigning a probability of 10^(-3) would mean being really, really, really, really sure that the hypothesis is wrong. To be well-calibrated, you would have to be able to make ten thousand similar judgments with similar strengths of evidence and only be wrong about ten times, and if you can do that, you're very good at this sort of thing.
Assigning 10^(-13) -- i.e., suggesting that you're so good that you can do this and only be wrong one in ten million million times -- is just obviously wrong.
So I was implying that the fact that the book suggests that this kind of number can be a plausible outcome means that it isn't a very good place to learn the art of making Bayesian probability estimates. To learn to make well-calibrated estimates, I should try to learn from people who stand a snowball's chance in hell of making such estimates themselves.
For an example from someone who has a claim to actually being good at this sort of thing, see Gwern's Who wrote the Death Note script?.
a claim to actually being good at this sort of thing
If I'm reading the chart on that page correctly, Gwern is extremely well calibrated. Is the accuracy row for each confidence column telling us what fraction of predictions Gwern assigned a given confidence to have been right? He's got 50% - 44%, 60% - 64%, 70% - 71%, 80% - 83%, 90% - 92%, and 100% - 96%. That's incredible!
Disclaimer: I have not read this book. I'm posting it in the expectation that others may enjoy it as much as I'm sure I would if I had time to read it myself.
This looks interesting as an extended worked example of Bayesian reasoning (the "scientific approach" of the title).
Edited to add:
There are many signs in the above block of text that this book is not up to Lesswrong standards. As gwern suggests, reading it should be done with an adversarial attitude.
I propose some more useful goals than finding someone for whom we can cheer loudly as a properly qualified member of our tribe: find worked examples that let you practice your art; find structured activities that will actually lead you to practice your art; try to critically assess arguments that use the tools we think powerful, then discuss your criticism on a forum like Lesswrong where your errors are likely to be discovered and your insights are likely to be rewarded (with tasty karma).