Reasoning has become a central paradigm for large language models (LLMs), consistently boosting accuracy across diverse benchmarks. Yet its suitability for precision-sensitive use remains unclear. We present the first systematic study of reasoning for classification tasks under strict low false positive rate (FPR) regimes. Our analysis covers two tasks—safety detection and hallucination detection—evaluated in both fine-tuned and zero-shot settings, using standard LLMs and Large Reasoning Models (LRMs). Our results reveal a clear trade-off: Think On (reasoning-augmented) generation improves overall accuracy, but performs poorly at the low-FPR thresholds essential for practical use. In contrast, Think Off (no reasoning during inference) dominates in these precision-sensitive regimes, with Think On surpassing only when higher FPRs are acceptable. In addition, we find token-based scoring substantially outperforms self-verbalized confidence for precision-sensitive deployments. Finally, a simple ensemble of the two modes recovers the strengths of each. Taken together, our findings position reasoning as a double-edged tool: beneficial for average accuracy, but often ill-suited for applications requiring strict precision.