Moving-target detection under strict sensing constraints is a recurring subproblem in surveillance, search-and-rescue, and autonomous robotics. We study a canonical one-dimensional finite grid in which a sensor probes one location per time step with binary observations while the target follows reflecting random-walk dynamics. The objective is to minimize the expected time to detection using transparent, training-free decision rules defined on the belief state of the target location. We compare two belief-driven heuristics with purely online implementation: a greedy rule that always probes the most probable location and a belief-proportional sampling (BPS, probability matching) rule that samples sensing locations according to the belief distribution (i.e., posterior probability of the target location). Repeated Monte Carlo simulations quantify the exploitation–exploration trade-off and provide a self-comparison between the two policies. Across tested grid sizes, the greedy policy consistently yields the shortest expected time to detection, improving by roughly 17–20% over BPS and uniform random probing in representative settings. BPS trades some average efficiency for stochastic exploration, which can be beneficial under model mismatch. This study provides an interpretable baseline and quantitative reference for extensions to noisy sensing and higher-dimensional search.