Abstract: Computing random-walk probabilities on graphs is the subject of extensive research in both graph theory and data mining research. However, existing work mainly focuses on static graphs, and cannot efficiently support dynamic weighted graphs, which are ubiquitous in real-world applications. We study the problem of computing random-walk probabilities on dynamic weighted graphs. We propose to use a sampling schema called coin flip sampling, rather than the more commonly adopted weighted sampling schema, for simulating random walks in dynamic weighted graphs. We demonstrate that simulations based on coin-flip sampling maintain the unbiasedness of the resulting random-walk probability approximations. Moreover, this approach allows us to simultaneously achieve a near-optimal query time complexity and an optimal O\left(1\right) update time overhead per edge insertion or deletion. This is a significant improvement over existing methods, which typically incur substantial sampling costs or rely on intricate auxiliary structures that are hard to maintain in a dynamic setting. We present both theoretical analysis and empirical evaluations to substantiate the superiority of our method on dynamic weighted graphs.