As robots become increasingly integrated into various industries, understanding how humans respond to robotic failures is critical. This study systematically examines trust dynamics and system design by analyzing human reactions to robot failures. We conducted a four-stage survey to explore how trust evolves throughout human-robot interactions. The first stage collected demographic data and initial trust levels. The second stage focused on preliminary expectations and perceptions of robotic capabilities. The third stage examined interaction details, including robot precision and error acknowledgment. Finally, the fourth stage assessed post-interaction perceptions, evaluating trust dynamics, forgiveness, and willingness to recommend robotic technologies. Results indicate that trust in robotic systems significantly increased when robots acknowledged their errors or limitations. Additionally, participants showed greater willingness to suggest robots for future tasks, highlighting the importance of direct engagement in shaping trust dynamics. These findings provide valuable insights for designing more transparent, responsive, and trustworthy robotic systems. By enhancing our understanding of human-robot interaction (HRI), this study contributes to the development of robotic technologies that foster greater public acceptance and adoption.