One mental model I apply to this and other similar problems is tracking what 'universe' you're in. In the flawed solution, the problem is that there's a univers

One mental model I apply to this and other similar problems is tracking what 'universe' you're in. In the flawed solution, the problem is that there's a universe in which the amounts are A and 2A and a universe in which the amounts are A and A/2, but they aren't the same universe, and the two possibilities will have different values for A and so you can't easily average A/2 from one universe with 2A from the other. (The usual term is sample space; it's just more fun to split and eliminate universes than states.)