



'''simple docstring'''



from itertools import product



def   __lowerCAmelCase					(							lowerCamelCase   :						int				,		lowerCamelCase   :						int					):






 '''simple docstring'''




 __lowerCAmelCase												=  sides_number
 __lowerCAmelCase												=  max_face_number * dice_number
 __lowerCAmelCase												=  [0] * (max_total + 1)

 __lowerCAmelCase												=  1
 __lowerCAmelCase												=  range(lowerCamelCase				,		max_face_number + 1					)
 for dice_numbers in product(lowerCamelCase				,		repeat=lowerCamelCase					):
  __lowerCAmelCase												=  sum(lowerCamelCase					)
  totals_frequencies[total] += 1

 return totals_frequencies



def   __lowerCAmelCase					(							):






 '''simple docstring'''




 __lowerCAmelCase												=  total_frequency_distribution(
     sides_number=4				,		dice_number=9					)
 __lowerCAmelCase												=  total_frequency_distribution(
     sides_number=6				,		dice_number=6					)

 __lowerCAmelCase												=  0
 __lowerCAmelCase												=  9
 __lowerCAmelCase												=  4 * 9
 __lowerCAmelCase												=  6
 for peter_total in range(lowerCamelCase				,		max_peter_total + 1					):
  peter_wins_count += peter_totals_frequencies[peter_total] * sum(
      colin_totals_frequencies[min_colin_total:peter_total]					)

 __lowerCAmelCase												=  (4**9) * (6**6)
 __lowerCAmelCase												=  peter_wins_count / total_games_number

 __lowerCAmelCase												=  round(lowerCamelCase				,		ndigits=7					)

 return rounded_peter_win_probability


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




'''simple docstring'''



# flake8: noqa
# Lint as: python3

from typing import Dict, List, Optional, Type

from .. import config
from ..utils import logging
from .formatting import (
    ArrowFormatter,
    CustomFormatter,
    Formatter,
    PandasFormatter,
    PythonFormatter,
    TensorFormatter,
    format_table,
    query_table,
)
from .np_formatter import NumpyFormatter


lowerCAmelCase	:		List[str]				=   logging.get_logger(__name__)

lowerCAmelCase	:		Dict[Optional[str], Type[Formatter]]				=   {}
lowerCAmelCase	:		Dict[Optional[str], str]				=   {}
lowerCAmelCase	:		Dict[Optional[str], Exception]				=   {}



def   __lowerCAmelCase					(							lowerCamelCase   :						type				,		lowerCamelCase   :						Optional[str]				,		lowerCamelCase   :						Optional[List[str]] = None				,		):






	'''simple docstring'''




	__lowerCAmelCase												=  aliases if aliases is not None else []
	if format_type in _FORMAT_TYPES:
		logger.warning(
		    f'''Overwriting format type \'{format_type}\' ({_FORMAT_TYPES[format_type].__name__} -> {formatter_cls.__name__})'''					)
	__lowerCAmelCase												=  formatter_cls
	for alias in set(aliases + [format_type]					):
		if alias in _FORMAT_TYPES_ALIASES:
			logger.warning(
			    f'''Overwriting format type alias \'{alias}\' ({_FORMAT_TYPES_ALIASES[alias]} -> {format_type})'''					)
		__lowerCAmelCase												=  format_type



def   __lowerCAmelCase					(							lowerCamelCase   :						Exception				,		lowerCamelCase   :						Optional[str]				,		lowerCamelCase   :						Optional[List[str]] = None					):






	'''simple docstring'''




	__lowerCAmelCase												=  aliases if aliases is not None else []
	for alias in set(aliases + [format_type]					):
		__lowerCAmelCase												=  unavailable_error


# Here we define all the available formatting functions that can be used by `Dataset.set_format`
_register_formatter(PythonFormatter, None, aliases=['''python'''])
_register_formatter(ArrowFormatter, '''arrow''', aliases=['''pa''', '''pyarrow'''])
_register_formatter(NumpyFormatter, '''numpy''', aliases=['''np'''])
_register_formatter(PandasFormatter, '''pandas''', aliases=['''pd'''])
_register_formatter(CustomFormatter, '''custom''')

if config.TORCH_AVAILABLE:
	from .torch_formatter import TorchFormatter

	_register_formatter(TorchFormatter, '''torch''', aliases=['''pt''', '''pytorch'''])
else:
	lowerCAmelCase	:		Optional[int]				=   ValueError('''PyTorch needs to be installed to be able to return PyTorch tensors.''')
	_register_unavailable_formatter(_torch_error, '''torch''', aliases=['''pt''', '''pytorch'''])

if config.TF_AVAILABLE:
	from .tf_formatter import TFFormatter

	_register_formatter(TFFormatter, '''tensorflow''', aliases=['''tf'''])
else:
	lowerCAmelCase	:		str				=   ValueError('''Tensorflow needs to be installed to be able to return Tensorflow tensors.''')
	_register_unavailable_formatter(_tf_error, '''tensorflow''', aliases=['''tf'''])

if config.JAX_AVAILABLE:
	from .jax_formatter import JaxFormatter

	_register_formatter(JaxFormatter, '''jax''', aliases=[])
else:
	lowerCAmelCase	:		Any				=   ValueError('''JAX needs to be installed to be able to return JAX arrays.''')
	_register_unavailable_formatter(_jax_error, '''jax''', aliases=[])



def   __lowerCAmelCase					(							lowerCamelCase   :						Optional[str]					):






	'''simple docstring'''




	if format_type in _FORMAT_TYPES_ALIASES:
		return _FORMAT_TYPES_ALIASES[format_type]
	else:
		return format_type




def   __lowerCAmelCase					(							lowerCamelCase   :						Optional[str]				,		**lowerCamelCase   :						Tuple					):






	'''simple docstring'''




	__lowerCAmelCase												=  get_format_type_from_alias(lowerCamelCase					)
	if format_type in _FORMAT_TYPES:
		return _FORMAT_TYPES[format_type](**lowerCamelCase					)
	if format_type in _FORMAT_TYPES_ALIASES_UNAVAILABLE:
		raise _FORMAT_TYPES_ALIASES_UNAVAILABLE[format_type]
	else:
		raise ValueError(
		    f'''Return type should be None or selected in {list(type for type in _FORMAT_TYPES.keys() if type != None					)}, but got \'{format_type}\''''					)