Satisficing heuristic search such as greedy best-first search (GBFS) suffers from local minima, regions where heuristic values are inaccurate and a good node has a worse heuristic value than other nodes. Search algorithms that incorporate exploration mechanisms in GBFS empirically reduce the search effort to solve difficult problems. Although some of these methods entirely ignore the guidance of a heuristic during their exploration phase, intuitively, a good heuristic should have some bound on its inaccuracy, and exploration mechanisms should exploit this bound. In this paper, we theoretically analyze what a good node is for satisficing heuristic search algorithms and show that the heuristic value of a good node has an upper bound if a heuristic satisfies a certain property. Then, we propose biased exploration mechanisms which select lower heuristic values with higher probabilities. In the experiments using synthetic graph search problems and classical planning benchmarks, we show that the biased exploration mechanisms can be useful. In particular, one of our methods, Softmin-Type(h), significantly outperforms other GBFS variants in classical planning and improves the performance of Type-LAMA, a state-of-the-art classical planner.