The Born Rule is a Feature of Superposition

Abstract: Finite probability theory is enriched by introducing the mathematical notion (no physics involved) of a superposition event $\Sigma S$--in addition to the usual discrete event $S$ (subset of the outcome space $U=\left( u_{1},...,u_{n}\right) $). Mathematically, the two types of events are distinguished using $n\times n$ density matrices. The density matrix $\rho\left( S\right) $ for a discrete event is diagonal and the density matrix $\rho\left( \Sigma S\right) $ is obtained as an outer product $\left\vert s\right\rangle \left\langle s\right\vert $ of a normalized vector $\left\vert s\right\rangle \in \mathbb{R}^{n}$. Probabilities are defined using density matrices as $\Pr\left( T|\rho\right) =\operatorname*{tr}\left[ P_{T}\rho\right] $ where $T\subseteq U$ and $P_{T}$ is the diagonal projection matrix with diagonal entries $\chi_{T}\left( u_{i}\right) $. Then for the singleton $\left\{ u_{i}\right\} \subseteq U$, the probability of the outcome $u_{i}$ conditioned by the \textit{superposition} event $\Sigma S$ is $\Pr\left( \left\{ u_{i}\right\} |\Sigma S\right) =\left\langle u_{i}|s\right\rangle ^{2}$, the Born Rule. Thus the Born Rule arises naturally from the mathematics of superposition when superposition events are added to ordinary finite probability theory. No further explanation is required when the mathematics uses $ \mathbb{C}^{n}$ instead of $\mathbb{R}^{n}$ except that the square $\left\langle u_{i}|s\right\rangle ^{2}$ is the absolute square $\left\vert \left\langle u_{i}|s\right\rangle \right\vert ^{2}$.