All the passage says is that if you believe the coin is unbiased, then you expect to see a roughly 50-50 split between heads and tails. If you expect to see 70:30 split of heads:tails, you ought to believe that the coin is so biased before you do the experiment. It looks trivial when applied to coins, but less so in other contexts. This is a statement about priors, not posteriors, hence the term "expectation". In Eliezer's example, if you are p% confident that an accused is a witch, then you should expect a definitive witch test to exonerate the accused (100-p)% of the time. If any outcome "confirms witchiness", then the test in question is not a test of witchiness.
Recently, I completed my first systematic read-through of the sequences. One of the biggest effects this had on me was considerably warming my attitude towards Bayesianism. Not long ago, if you'd asked me my opinion of Bayesianism, I'd probably have said something like, "Bayes' theorem is all well and good when you know what numbers to plug in, but all too often you don't."
Now I realize that that objection is based on a misunderstanding of Bayesianism, or at least Bayesianism-as-advocated-by-Eliezer-Yudkowsky. "When (Not) To Use Probabilities" is all about this issue, but a cleaner expression of Eliezer's true view may be this quote from "Beautiful Probability":
The practical upshot of seeing Bayesianism as an ideal to be approximated, I think, is this: you should avoid engaging in any reasoning that's demonstrably nonsensical in Bayesian terms. Furthermore, Bayesian reasoning can be fruitfully mined for heuristics that are useful in the real world. That's an idea that actually has real-world applications for human beings, hence the title of this post, "Bayesianism for Humans."
Here's my attempt to make an initial list of more directly applicable corollaries to Bayesianism. Many of these corollaries are non-obvious, yet eminently sensible once you think about them, which I think makes for a far better argument for Bayesianism than Dutch Book-type arguments with little real-world relevance. Most (but not all) of the links are to posts within the sequences, which hopefully will allow this post to double as a decent introductory guide to the parts of the sequences that explain Bayesianism.