


from itertools import product
def     __lowerCamelCase						(			UpperCAmelCase_						:		int			,  UpperCAmelCase_						:		int						):




      """simple docstring"""





      a						:str										=	sides_number
      a						:Any										=	max_face_number * dice_number
      a						:str										=	[0] * (max_total + 1)

      a						:Union[str, Any]										=	1
      a						:Tuple										=	range(UpperCAmelCase_			,  max_face_number + 1						)
      for dice_numbers in product(UpperCAmelCase_			,  repeat=UpperCAmelCase_						):
            a						:Any										=	sum(UpperCAmelCase_						)
            totals_frequencies[total] += 1

      return totals_frequencies
def     __lowerCamelCase						(			):




      """simple docstring"""





      a						:Union[str, Any]										=	total_frequency_distribution(
          sides_number=4			,  dice_number=9						)
      a						:Union[str, Any]										=	total_frequency_distribution(
          sides_number=6			,  dice_number=6						)

      a						:Union[str, Any]										=	0
      a						:Any										=	9
      a						:List[Any]										=	4 * 9
      a						:Optional[int]										=	6
      for peter_total in range(UpperCAmelCase_			,  max_peter_total + 1						):
            peter_wins_count += peter_totals_frequencies[peter_total] * sum(
                colin_totals_frequencies[min_colin_total:peter_total]						)

      a						:str										=	(4**9) * (6**6)
      a						:Optional[Any]										=	peter_wins_count / total_games_number

      a						:Tuple										=	round(UpperCAmelCase_			,  ndigits=7						)

      return rounded_peter_win_probability


if __name__ == "__main__":
     print(F"""{solution() = }""")









from . import __version__

# Backward compatibility imports, to make sure all those objects can be found in file_utils
from .utils import (
    CLOUDFRONT_DISTRIB_PREFIX,
    CONFIG_NAME,
    DISABLE_TELEMETRY,
    DUMMY_INPUTS,
    DUMMY_MASK,
    ENV_VARS_TRUE_AND_AUTO_VALUES,
    ENV_VARS_TRUE_VALUES,
    FEATURE_EXTRACTOR_NAME,
    FLAX_WEIGHTS_NAME,
    HF_MODULES_CACHE,
    HUGGINGFACE_CO_PREFIX,
    HUGGINGFACE_CO_RESOLVE_ENDPOINT,
    MODEL_CARD_NAME,
    MULTIPLE_CHOICE_DUMMY_INPUTS,
    PYTORCH_PRETRAINED_BERT_CACHE,
    PYTORCH_TRANSFORMERS_CACHE,
    S3_BUCKET_PREFIX,
    SENTENCEPIECE_UNDERLINE,
    SPIECE_UNDERLINE,
    TF2_WEIGHTS_NAME,
    TF_WEIGHTS_NAME,
    TORCH_FX_REQUIRED_VERSION,
    TRANSFORMERS_CACHE,
    TRANSFORMERS_DYNAMIC_MODULE_NAME,
    USE_JAX,
    USE_TF,
    USE_TORCH,
    WEIGHTS_INDEX_NAME,
    WEIGHTS_NAME,
    ContextManagers,
    DummyObject,
    EntryNotFoundError,
    ExplicitEnum,
    ModelOutput,
    PaddingStrategy,
    PushToHubMixin,
    RepositoryNotFoundError,
    RevisionNotFoundError,
    TensorType,
    _LazyModule,
    add_code_sample_docstrings,
    add_end_docstrings,
    add_start_docstrings,
    add_start_docstrings_to_model_forward,
    cached_property,
    copy_func,
    default_cache_path,
    define_sagemaker_information,
    get_cached_models,
    get_file_from_repo,
    get_full_repo_name,
    get_torch_version,
    has_file,
    http_user_agent,
    is_apex_available,
    is_bsa_available,
    is_coloredlogs_available,
    is_datasets_available,
    is_detectrona_available,
    is_faiss_available,
    is_flax_available,
    is_ftfy_available,
    is_in_notebook,
    is_ipex_available,
    is_librosa_available,
    is_offline_mode,
    is_onnx_available,
    is_pandas_available,
    is_phonemizer_available,
    is_protobuf_available,
    is_psutil_available,
    is_pyanvml_available,
    is_pyctcdecode_available,
    is_pytesseract_available,
    is_pytorch_quantization_available,
    is_rjieba_available,
    is_sagemaker_dp_enabled,
    is_sagemaker_mp_enabled,
    is_scipy_available,
    is_sentencepiece_available,
    is_seqio_available,
    is_sklearn_available,
    is_soundfile_availble,
    is_spacy_available,
    is_speech_available,
    is_tensor,
    is_tensorflow_probability_available,
    is_tfaonnx_available,
    is_tf_available,
    is_timm_available,
    is_tokenizers_available,
    is_torch_available,
    is_torch_bfaa_available,
    is_torch_cuda_available,
    is_torch_fx_available,
    is_torch_fx_proxy,
    is_torch_mps_available,
    is_torch_tfaa_available,
    is_torch_tpu_available,
    is_torchaudio_available,
    is_training_run_on_sagemaker,
    is_vision_available,
    replace_return_docstrings,
    requires_backends,
    to_numpy,
    to_py_obj,
    torch_only_method,
)





