






def      UpperCAmelCase						( lowerCAmelCase__					,     lowerCAmelCase__ = 1000	):


		'''simple docstring'''

		if (not isinstance(__lowercase					,     __lowercase	)) or (digit_position <= 0):
				raise ValueError("Digit position must be a positive integer"	)
		elif (not isinstance(__lowercase					,     __lowercase	)) or (precision < 0):
				raise ValueError("Precision must be a nonnegative integer"	)

		# compute an approximation of (16 ** (n - 1)) * pi whose fractional part is mostly
		# accurate
		__A    =							(
		    4 * _subsum(__lowercase					,     1					,     __lowercase	)
		    - 2 * _subsum(__lowercase					,     4					,     __lowercase	)
		    - _subsum(__lowercase					,     5					,     __lowercase	)
		    - _subsum(__lowercase					,     6					,     __lowercase	)
		)

		# return the first hex digit of the fractional part of the result
		return hex(int((sum_result % 1) * 16	)	)[2:]



def      UpperCAmelCase						( lowerCAmelCase__					,     lowerCAmelCase__					,     lowerCAmelCase__	):


		'''simple docstring'''

		__A    =							0.0
		for sum_index in range(digit_pos_to_extract + precision	):
				__A    =							8 * sum_index + denominator_addend
				if sum_index < digit_pos_to_extract:
						# if the exponential term is an integer and we mod it by the denominator
						# before dividing, only the integer part of the sum will change;
						# the fractional part will not
						__A    =							pow(
						    16					,     digit_pos_to_extract - 1 - sum_index					,     __lowercase	)
				else:
						__A    =							pow(16					,     digit_pos_to_extract - 1 - sum_index	)
				total += exponential_term / denominator
		return total


if __name__ == "__main__":
					import doctest

					doctest.testmod()







'''simple docstring'''


from __future__ import annotations

import random






class 						lowerCAmelCase_	:







  '''simple docstring'''

  UpperCamelCase_			:      Optional[int]										=				6



  def __init__( self				:							Dict	)			->		Dict:







       '''simple docstring'''



       A:		Union[str, Any]          =		list(range(1					,  Dice.NUM_SIDES + 1	)	)




  def 					_snake_case     ( self				:							Optional[Any]	)			->		Any:







       '''simple docstring'''



       return random.choice(self.sides	)






def 					SCREAMING_SNAKE_CASE(   __lowercase		,				__lowercase = 2	)			-> list[float]:
     A:		Union[str, Any]          =		[Dice() for i in range(__lowercase	)]
     A:		Union[str, Any]          =		[0] * (len(__lowercase	) * Dice.NUM_SIDES + 1)
     for _ in range(__lowercase	):
          count_of_sum[sum(dice.roll() for dice in dices	)] += 1
     A:		Dict          =		[round((count * 1_0_0) / num_throws		,				2	) for count in count_of_sum]
     return probability[num_dice:]  # remove probability of sums that never appear


if __name__ == "__main__":
     import doctest

     doctest.testmod()






