Sparse Knowledge Graphs (KGs) are commonly encountered in real-world applications, where knowledge is often incomplete or limited. Sparse KG reasoning, the task of inferring missing knowledge over sparse KGs, is inherently challenging due to the scarcity of knowledge and the difficulty of capturing relational patterns in sparse scenarios. Among all sparse KG reasoning methods, path-based ones have attracted plenty of attention due to their interpretability. Existing path-based methods typically rely on computationally intensive random walks to collect paths, producing paths of variable quality. Additionally, these methods fail to leverage the structured nature of graphs by treating paths independently. To address these shortcomings, we propose a Structural and Probabilistic framework named StruProKGR, tailored for efficient and interpretable reasoning on sparse KGs. StruProKGR utilizes a distance-guided path collection mechanism to significantly reduce computational costs while exploring more relevant paths. It further enhances the reasoning process by incorporating structural information through probabilistic path aggregation, which prioritizes paths that reinforce each other. Extensive experiments on five sparse KG reasoning benchmarks reveal that StruProKGR surpasses existing path-based methods in both effectiveness and efficiency, providing an effective, efficient, and interpretable solution for sparse KG reasoning.