In real path selection decisions, besides the cost of the path which is the major decision factor, the chance of the path connection is also significant. A path which has low weight but low chance of existence is not a good choice for decision making. In order to make network optimization more in line with the actual situation, this paper studies one model of it on a network whose existence and weight of edges are both indeterminate. The aim of this paper is to obtain a spanning tree which satisfies the connectivity constraint and minimizes the total weight as well. The study uses conditional uncertain variable to discribe the relationship between these two aspects of uncertain variables. On this basis, three different models of the UMST problem are suggested, and the equivalence relation between the MST model of uncertain networks with uncertain edge existence and weight and the MST of related deterministic networks is found. Examples of experiments with specific numerical values are provided to verify the variation of the network when two variables are considered simultaneously.