Even after looking at those numbers, it still felt wrong to me. It eventually occurred to me that this was because my intuition was telling me the total should

Even after looking at those numbers, it still felt wrong to me. It eventually occurred to me that this was because my intuition was telling me the total should add up to A. And it doesn't; it adds up to 9/8 A. It took me a while to realize that my intuition was wrong, and that there's no reason the total should be A. The thing that helped me reconcile this was realizing that the A in the two equations are different values; then replacing them. (0.5⋅(0.5⋅A + 0.5⋅2.A)) + (0.5⋅(0.5⋅A + 0.5⋅0.5⋅A)) = 9/8 A ^ but the A in the left grouping (where it's 100) is different than the A in the right grouping (where it's 200). Replacing the As with their actual values 0.5⋅(0.5⋅100+0.5⋅2⋅100)+0.5⋅(0.5⋅200+0.5⋅0.5⋅200) = 150 And, since one envelope has 100 and the other has 200, an expected outcome across both envelopes is, as calculated, 150.