In legal matters, text classification models are most often used to filter through large datasets in search of documents that meet certain pre-selected criteria like relevance to a certain subject matter, such as legally privileged communications and attorney-directed documents. In this context, large language models have demonstrated strong performance. This paper presents an empirical study investigating the role of randomness in LLM-based classification for attorney-client privileged document detection, focusing on four key dimensions: (1) the effectiveness of LLMs in identifying legally privileged documents, (2) the influence of randomness control parameters on classification outputs, (3) their impact on overall classification performance, and (4) a methodology for leveraging randomness to enhance accuracy. Experimental results showed that LLMs can identify privileged documents effectively, randomness control parameters have minimal impact on classification performance, and importantly, our developed methodology for leveraging randomness can have a significant impact on improving accuracy. Notably, this methodology that leverages randomness could also enhance a corporation's confidence in an LLM's output when incorporated into its sanctions-compliance processes. As organizations increasingly rely on LLMs to augment compliance workflows, reducing output variability helps build internal and regulatory confidence in LLM-derived sanctions-screening decisions.