# Flipping Against All Odds: Reducing LLM Coin Flip Bias via Verbalized Rejection Sampling

arXiv: https://arxiv.org/abs/2506.09998 · HTML: https://mlsanity.com/paper/2506.09998/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 21 references that are in the map.
- First posted on arXiv 11 Jun 2025.

## Builds on (most cited first, 13 of 13)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Benchmarking Distributional Alignment of Large Language Models](https://mlsanity.com/paper/2411.05403/) (2024)
- [LLM Processes: Numerical Predictive Distributions Conditioned on Natural Language](https://mlsanity.com/paper/2405.12856/) (2024)
- [Specializing Large Language Models to Simulate Survey Response Distributions for Global Populations](https://mlsanity.com/paper/2502.07068/) (2025)
- [Heads, Tails, and AI Fails: LLMs, Randomness, and Human Judgments](https://mlsanity.com/paper/2406.00092/) (2024)
- [Enough Coin Flips Can Make LLMs Act Bayesian](https://mlsanity.com/paper/2503.04722/) (2025)
- [Verbalized Machine Learning: Revisiting Machine Learning with Language Models](https://mlsanity.com/paper/2406.04344/) (2024)
- [Do LLMs Play Dice? Exploring Probability Distribution Sampling in Large Language Models for Behavioral Simulation](https://mlsanity.com/paper/2404.09043/) (2024)
- [Vocabulary for Universal Approximation: A Linguistic Perspective of Mapping Compositions](https://mlsanity.com/paper/2305.12205/) (2023)
- [Evaluating Binary Decision Biases in Large Language Models: Implications for Fair Agent-Based Financial Simulations](https://mlsanity.com/paper/2501.16356/) (2025)

## Built on by (most cited first, 7 of 7)

- [The Illusion of Stochasticity in LLMs](https://mlsanity.com/paper/2604.06543/) (2026)
- [Large Language Models Are Bad Dice Players: LLMs Struggle to Generate Random Numbers from Statistical Distributions](https://mlsanity.com/paper/2601.05414/) (2026)
- [Probabilistic Calibration Is a Trainable Capability in Language Models](https://mlsanity.com/paper/2605.11845/) (2026)
- [Instruction-Tuned Language Models Cannot Sample from Distributions They Can Describe](https://mlsanity.com/paper/2607.25292/) (2026)
- [Structured Inference with Large Language Gibbs](https://mlsanity.com/paper/2606.19264/) (2026)
- [Sample What You Say: Aligning Language Models to Sample the Distributions They State](https://mlsanity.com/paper/2609.34929/) (2026)
- [Beyond Answer Confidence: A Controlled Audit of Self-Knowledge in a Black-Box Decision Model](https://mlsanity.com/paper/2610.01006/) (2026)

## Abstract (arXiv)

Large language models (LLMs) can often accurately describe probability distributions using natural language, yet they still struggle to generate faithful samples from them. This mismatch limits their use in tasks requiring reliable stochasticity, such as Monte Carlo methods, agent-based simulations, and randomized decision-making. We investigate this gap between knowledge and sampling in the context of Bernoulli distributions. We introduce Verbalized Rejection Sampling (VRS), a natural-language adaptation of classical rejection sampling that prompts the LLM to reason about and accept or reject proposed samples. Despite relying on the same Bernoulli mechanism internally, VRS substantially reduces sampling bias across models. We provide theoretical analysis showing that, under mild assumptions, VRS improves over direct sampling, with gains attributable to both the algorithm and prompt design. More broadly, our results show how classical probabilistic tools can be verbalized and embedded into LLM workflows to improve reliability, without requiring access to model internals or heavy prompt engineering.