Question: In Dirac's Principles of QM following is stated:

$$
\langle x | A + B | x \rangle = \langle x | A | x \rangle + \langle x | B |x \rangle
$$

but

$$
\langle x | AB | x \rangle \ne \langle x | A | x \rangle \langle x | B | x \rangle
$$

and so $\langle x|A|x \rangle$ is not exact but average value of observable A otherwise in second relation both sides would have to be equal.
I don't understand the second relation. Shouldn't both side be equal like this,
$\langle x|AB|x \rangle = \langle x|A(B|x \rangle) = b \langle x|A|x \rangle = ba \langle x|x \rangle = ba = \langle x|A|x \rangle \langle x|B|x \rangle$ , where $a$ and $b$ are corresponding eigenvalues. What is wrong here?

Edit: It is embarrassing. I indeed was thinking $|x\rangle$ was eigenvector of both $A$ and $B$ out of sleep deprivation I suppose which I only realised this morning. So anyway I am going to ask moderator to delete this question.

Answer: 
When submerged, the coin displaces as much water as it has volume (logical).

When floating on the box, the coin displaces as much water as corresponds to its weight.

As metal has a higher density than water, it means that the coin in the box displaces more water than when the coin is submerged.
The key is that the box can bob up and down. If both were completely submerged to begin with, then you'd be right: no rearranging of them underwater will cause some sort of strange change in displacement. But because the box is floating, so that *not all of it* is displacing water, the amount of water it displaces can change if it moves up or down, thus bringing relatively more or less box into the water.

When your coin falls off, the box bobs slightly up, which *reduces* the amount of water it displaces.

Hence, the water level falls.

(Note: actually, you need a bit more quantitative analysis to see it *will* fall, because technically we need to also account the change in displacement of the water when the coin enters it now which will compete against the bobbing to see that the latter will, in fact, win, but the point here is to show where and how, *conceptually*, the fall in water level can arise despite the seemingly "added displacement".)