Well, of course Bayesianism is your friend here. Probability theory elegantly supersedes the qualitative concepts of "knowledge", "belief" and "justification" and, together with an understanding of heuristics and biases, nicely dissolves Gettier problems, so that we can safely call "knowledge" any assignment of high probability to a proposition that turns out to be true.

I agree that, with regard to my own knowledge, I should just determine the probability that I assign to a proposition P. Once I conclude that P has a high probability of being true, why should I care whether, in addition, I "know" P in some sense?

Nonetheless, if I had to develop a coherent concept of "knowledge", I don't think that I'd go with "'knowledge' [is] any assignment of high probability to a proposition that turns out to be true." The crucial question is, who is assigning the probability? If it's my assignment, then, as I said, I agree that, for me, the question about knowledge dissolves. (More generally, the question dissolves if the assignment was made according to my prior and my cognitive strategies.)

But Getteir problems are usually about some third person's knowledge. When do you say that they know something? Suppose that, by your lights, they have a hopelessly screwed-up prior — say, an anti-Laplacian prior. So, they assign high probability to all sorts of stupid things for no good reason. Nonetheless, they have enough beliefs so that there are some things to which they assign high probability that turn out to be true. Would you really want to say that they "know" those things that just happen to be true?

That is essentially what was going on in my example with Mr. X's book. There, I'm the third person. I have the stupid prior that says that everything in B is true and everything not in B is false. Now, you know that Mr. X is constitutionally compelled to write truthfully about his mother's socks. So you know that reading B will legitimately entangle my beliefs with reality on that one solitary subject. But I don't know that fact about Mr. X. I just believe everything in B. You know that my cognitive strategy will give me reliable knowledge on this one subject. But, intuitively, my epistemic state seems so screw-up that you shouldn't say that I know anything, even though I got this one thing right.

ETA: Gah. This is what I meant by "down the rabbit-hole". These kinds of conversations are just too fun :). I look forward to your reply, but it will be at least a day before I reply in turn.

ETA: Okay, just one more thing. I just wanted to say that I agree with your approach to the original Gettier problem with the coins.