 N-box pre- and post-selection paradox

For a uniform prestate in $N+1$ dimensions and poststate proportional to ${\sum }_{i=1}^N|i⟩-(N-1)|N+1⟩$, each binary question $P_i$ versus $I-P_i$ has conditional probability one for $i\le N$. A fully resolved quantum measurement instead assigns $1/D$ to each of the first $N$ outcomes and $(N-1{)}^2/D$ to the last. The context dependence of pre- and post-selected measurements explains why these alternative certainties do not imply simultaneous properties.