Simpler: probability is about finding the relative size of a shape.
When you want to know A|B, that means you want to know what fraction of B is taken up by the intersection A-AND-B.
Well, say you happen to know B|A, or how much A-AND-B takes up inside of A. And you also know the size of A relative to the entire page, and B relative to the same.
Then, you can compute the "absolute" size of A-AND-B by multiplying the fraction of A that A-AND-B takes up by the fraction of the page that A takes up.
Now that you know the "absolute" size of A-AND-B, you can freely compare it to lots of other shapes. In this case, you can find A|B as simply as computing A-AND-B/B.
This is identical to your explanation, except I find it helpful to focus less about chance and expectation, and to think more about geometry and sets of outcomes.
Note that my notion of "absolute" size is still just relative. You could measure size relative to any other shape, but "size relative to the entire page" just happens to be a measure that we often know.
That's a good intuition, but I think the author avoided that for some important foundational reasons.
There are two main ways of developing probability theory.
The first way is to use measure theory. This means that probability is literally defined as an area or volume, and leads to the picture you describe. However, it forces you to think of probability as picking from a set of possiblities, and it also means you can arbitrarily carve up a shape or jam infinitely many of them together. You also need to be quite careful, as you may be taking the area of some rather weird shapes.
The second way is to assign "plausibilities" to various propositions, and then use them following a few common-sense rules. This assumes a lot less, but Richard Cox proved it gives the same result as the previous method when they both apply. However, you can now talk about the probability of rain without needing to be able to decompose it into statements about the trajectory of clouds. This also lets you talk about the probability the sun will rise tomorrow or the probability I am typing this from a hot tub without thinking about possible worlds or alternate histories.
The author is a huge proponent of the latter approach, and thus likely tried to avoid talking about shapes or regions.
In high school, you probably learned the classical definition of probability, where, if I say that something happens with probability A/B, it means that, if I do an event infinitely often, on average it will happen A out of B times. This turns out to be too fragile. I can make a die where the probability of rolling an odd number is 1/2 and the probability of rolling a number below 3 is 1/3, but the probability of rolling a 1 is undefined!
I know that there's some subtlety to correctly getting meaning out of probability, and I vaguely remember having similar doubts about my approach because of this.
That said, this simple model didn't fail to explain any of the concepts I learned in elementary discrete and continuous probability. It was especially useful for understanding the latter.
I don't have a lot of brain-RAM, and I find that I get confused the second that I stop focusing on "sets of outcomes". Also, for me, attempts to use "common-sense rules" about expectation seem to fail quickest of all.
When you want to know A|B, that means you want to know what fraction of B is taken up by the intersection A-AND-B.
Well, say you happen to know B|A, or how much A-AND-B takes up inside of A. And you also know the size of A relative to the entire page, and B relative to the same.
Then, you can compute the "absolute" size of A-AND-B by multiplying the fraction of A that A-AND-B takes up by the fraction of the page that A takes up.
Now that you know the "absolute" size of A-AND-B, you can freely compare it to lots of other shapes. In this case, you can find A|B as simply as computing A-AND-B/B.
This is identical to your explanation, except I find it helpful to focus less about chance and expectation, and to think more about geometry and sets of outcomes.
Note that my notion of "absolute" size is still just relative. You could measure size relative to any other shape, but "size relative to the entire page" just happens to be a measure that we often know.