For a connected simple graph G whose starting point is v, by using equal probability to move to the next edge constantly, a sequential walk whose points and edges can be reused is formed. When going through the starting point once, it is called a special closed walk. This study aims at investigating the expected values of edges in these special closed walks. Considering whether an immediate reversal is possible during the movement, I classify the problem into two types of expected values. By utilizing the concept of matrix solving for a system of equations, it provides an algorithm to find expected values. Additionally, I explore the relations between these two types of expected values. I also characterize expected values by using edge number and vertex degree. I also attempt to modify the principle of equal probabilities of selection of edges and generalize it, investigating the special properties of expected values and characterizing the necessary and sufficient conditions.