In this work we develop a method for analyzing maximum weight selections in matroids with random element weights, especially exponentially distributed weights. We use the structure of the matroid dual to transform matroid maximization into an equivalent minimization task. We model sample paths of the greedy minimization scheme using a Markov process, and thus solve the original maximization problem. The distribution of the weight of the optimal basic element and moments are found, as well as the probability that a given basic element is optimal. We also derive criticality indices for each ground set element, giving the probability that an element is a member of the optimal solution. We give examples using spanning trees and scheduling problems, each example being a new result in stochastic combinatorial optimization.