Dagger categories (a.k.a.  \(*\) -categories) can be seen as categories with a notion of “transpose”, generalizing the transposition of matrices in linear algebra. This allows us to extend the ideas of orthogonality and orthogonal projector from Euclidean geometry and Hilbert space theory to a much more general and abstract context. By means of a dagger category of probability spaces and transport plans, we show that this abstract notion of orthogonality can model exactly independence and conditional independence of random variables. Moreover, orthogonal projectors correspond exactly to conditioning, giving a unified description of “observations” for both quantum and classical experiments.