Do Large Language Models Really Understand Probability? — A Benchmark Study on LLM Probabilistic Reasoning Capabilities

This study conducts a benchmark test on the discrete probabilistic reasoning capabilities of large language models (LLMs). The results show: LLMs achieve an average accuracy of 96% on regular probability problems, but drop sharply to 59% when facing counterintuitive problems; models are extremely sensitive to prompt wording, with wording changes leading to a performance drop of over 20%; chain-of-thought (CoT) prompts have limited improvement on counterintuitive problems. The research source is the paper "How reliable are LLMs when it comes to playing dice?" published on arXiv on June 5, 2026 (link: http://arxiv.org/abs/2606.07515v1), which reminds that LLMs should be used cautiously in high-risk decision-making fields.

LLMs have made significant progress in fields like mathematical reasoning and code generation, even handling problems at the level of international Olympiads. But the core question is: Does the model understand concepts or imitate patterns? Probabilistic reasoning is a "trap area" for human cognition (e.g., Monty Hall problem, birthday paradox). If LLMs are to become reliable reasoning tools, they need to remain robust in the face of these cognitive traps, which is the motivation for this study.

The study evaluates 8 advanced LLMs and constructs two datasets: 1. Standard exercise set (regular discrete probability problems); 2. Counterintuitive exercise set (problems that trigger heuristic errors). Each model is tested under two conditions: zero-shot and chain-of-thought (CoT) prompts, to separate model capabilities from prompt gains.

Models achieve an average accuracy of 96% on standard problems, but drop sharply to 59% on counterintuitive ones—this phenomenon exists across all models. This indicates that LLM probabilistic reasoning is "shallow": they can handle regular problems but lack true probabilistic intuition and struggle to resist intuitive misleading.

The study finds that models are extremely sensitive to prompts: 1. Token bias effect: Problems that are mathematically equivalent but have different wording lead to a performance drop of over 20%; 2. Misleading prompts: Embedding guidance like "Intuitively you might think..." leads to a performance drop of up to 34%. This shows that models rely on specific vocabulary and sentence patterns rather than abstract conceptual reasoning.

Chain-of-thought prompts significantly improve performance on standard problems, but have limited improvement on counterintuitive ones. This suggests that CoT mainly helps organize known information rather than correct reasoning biases—when problems require overcoming intuitive misleading, just "thinking step by step" cannot solve the fundamental issue.

Practical implications: 1. High-risk fields (medical, finance, etc.) need human supervision and verification; 2. Robust prompt templates and input validation mechanisms need to be developed. Future directions: 1. Evaluation frameworks need to include more counterintuitive/adversarial tests; 2. Research on how to endow LLMs with true probabilistic intuition and improve prompt robustness.

Although LLMs perform amazingly on multiple tasks, this study shows that they have deep flaws in discrete probabilistic reasoning capabilities and are still far from being reliable reasoning engines. Before deploying AI systems, we need to clearly understand their capability boundaries and establish safety guarantee mechanisms.