🎲 Large Language Models Are Bad Dice Players

If you use this code, the released prompts, or the data in your research, please cite our paper:

@inproceedings{zhao-etal-2026-large,
    title = "Large Language Models Are Bad Dice Players: {LLM}s Struggle to Generate Random Numbers from Statistical Distributions",
    author = "Zhao, Minda and Du, Yilun and Wang, Mengyu",
    booktitle = "Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)",
    month = jul,
    year = "2026",
    address = "San Diego, California, United States",
    publisher = "Association for Computational Linguistics",
    url = "https://aclanthology.org/2026.acl-long.1051/",
    doi = "10.18653/v1/2026.acl-long.1051",
    pages = "22942--22959"
}

Machine-readable citation metadata are also available in CITATION.cff.

Can a large language model natively sample from a specified probability distribution without calling an external numerical tool? We audit this capability at scale across 11 frontier LLMs, 15 distributions, and 1,000 samples per configuration.

The benchmark separates two settings:

• Batch Generation: one response contains all 1,000 requested samples.
• Independent Requests: 1,000 stateless calls each produce one sample.

The released benchmark uses Wasserstein-1 distance for distributional fidelity, two-sample Kolmogorov–Smirnov tests for continuous distributions, and chi-square goodness-of-fit tests for discrete distributions, with alpha = 0.01.

• Batch generation achieves only modest validity: the median model passes 7% of the 15 distributions, and the strongest model passes 40%.
• Independent requests nearly collapse: 10 of 11 models pass none of the distributions.
• Sampling fidelity worsens as distributional complexity and the sampling horizon increase.
• The same failure propagates into applications: all evaluated models show significant answer-position bias in MCQ generation and violate target distributions in attribute-constrained text-to-image prompt generation.
• For applications requiring statistical guarantees, use an external, validated sampler rather than relying on native LLM sampling.

A pass means that the corresponding statistical test does not reject the target distribution at alpha = 0.01. See Tables 2–3 and 9 in the paper for Wasserstein-1 distances and tier-level results.

Each cell reports Wasserstein-1 distance (lower is better); circles mark configurations that pass the corresponding statistical test at alpha = 0.01. Colors use a logarithmic scale.

For the downstream medical MCQ experiment, each model generated N = 1000 questions under an explicit instruction to distribute correct answers uniformly across A/B/C/D. All six models still deviated significantly from the 25% target (p < 0.001).

The bars report the percentage of correct answers assigned to each option; the red crosses mark rejection of the uniform target. GPT-OSS-120B shows the strongest skew toward option C (54.6%), while GPT-4o favors option B (46.8%). See Table 4 in the paper.

This repository includes the code, exact prompt templates, raw model outputs, reference samples, and processed summaries used by the release:

configs/
  distributions.json        distribution definitions and paper parameters
  protocols.json            batch, independent, and sample-size settings
  evaluation.json           tests, metrics, alpha, and reference seed
  models.template.json      model registry without credentials
  downstream_tasks.json     MCQ and attribute-task settings

prompts/
  batch/                    15 batch-generation prompts
  independent/              15 independent-request prompts
  downstream/               MCQ and attribute-generation prompts

data/
  raw_results/
    batch/                  released batch outputs
    independent/            released independent-request outputs
  reference/                aligned NumPy/SciPy reference samples
  downstream/
    mcq/raw_outputs/        released MCQ outputs
    attributes/joint/       released joint-attribute outputs
    attributes/independent/ released single-attribute outputs
  processed/
    recomputed_main_results.json
    downstream/downstream_results.json

manifests/                  source and prompt provenance manifests
assets/                     README figures
docs/paper_contract.md      canonical reproducibility settings
src/                        generation, parsing, metrics, and release checks
scripts/validate_release.sh one-command local validation

For a quick look at the published results, start with:

• data/processed/recomputed_main_results.json for the main batch and independent benchmark.
• data/processed/downstream/downstream_results.json for MCQ and attribute-constrained generation.
• docs/paper_contract.md for the canonical experiment settings.

git clone https://github.com/Mininda/LLM_Bad_Dice_Player.git
cd LLM_Bad_Dice_Player

python -m venv .venv
source .venv/bin/activate
python -m pip install --upgrade pip
pip install -r requirements.txt

bash scripts/validate_release.sh

The validation command checks Python syntax, required release files, prompt counts, accidental secret patterns, processed-result structure, and the headline pass rates reported in the paper. It does not make API calls.

Generation requires credentials for the providers you use. The scripts read environment-variable names from configs/models.template.json; no credentials are stored in the repository.

export OPENAI_API_KEY=...
export DEEPINFRA_API_KEY=...
export GEMINI_API_KEY=...

Provider model identifiers and availability may change over time. Check configs/models.template.json before launching a new run and update only the relevant api_model or base_url entry when necessary.

Batch generation:

python src/generate_samples.py \
  --protocol batch \
  --model GPT-4o \
  --distribution Gaussian \
  --n-samples 1000 \
  --output-dir outputs/main

Independent requests:

python src/generate_samples.py \
  --protocol independent \
  --model GPT-4o \
  --distribution Uniform \
  --n-samples 1000 \
  --output-dir outputs/main

Valid distribution keys are listed in configs/distributions.json. Valid model display names are listed in configs/models.template.json.

MCQ generation:

python src/generate_downstream.py \
  --task mcq \
  --model GPT-4o \
  --n-samples 1000 \
  --output-dir outputs/downstream

Joint attribute-constrained prompt generation:

python src/generate_downstream.py \
  --task joint_attribute \
  --model DeepSeek-V3.2 \
  --n-samples 1000 \
  --output-dir outputs/downstream

Single-attribute follow-up:

python src/generate_downstream.py \
  --task independent_height \
  --model GPT-OSS \
  --n-samples 1000 \
  --output-dir outputs/downstream

Other supported tasks are independent_gender, independent_race, and independent_color. The downstream runner uses 32 concurrent workers by default; reduce this with --max-workers if required by your provider's rate limits.

All main experiments use N = 1000, temperature = 1.0, and top_p = 1.0. Reference samples use a fixed seed of 42. The exact parameters and evaluation choices live in configs/ rather than being duplicated in the runners.

This repository is released under the Apache License 2.0.