Bundles of Probability Schemes

We study finite probability theory through a category of finite probability schemes and probability-preserving maps, called \emph{bundles}. A bundle simultaneously records a quotient of a sample space, an algebra of random variables, and the family of conditional schemes over the quotient. The two natural linear functors associated with a bundle give a compact construction of conditional expectation and explain its projection properties. Within this framework we recover the laws of total expectation, variance, and covariance, the weak law of large numbers, and the variance decomposition behind simple linear regression. Fiber products then encode conditional independence and discrete-time Markov chains.