

"""simple docstring"""






from itertools import product

def 		lowerCamelCase__		(					_lowerCamelCase ,   _lowerCamelCase   ):

    '''simple docstring'''



    _lowerCAmelCase       :   int  =    sides_number
    _lowerCAmelCase       :   Optional[Any]  =    max_face_number * dice_number
    _lowerCAmelCase       :   List[str]  =    [0] * (max_total + 1)

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

    return totals_frequencies

def 		lowerCamelCase__		(					):

    '''simple docstring'''



    _lowerCAmelCase       :   Dict  =    total_frequency_distribution(
        sides_number=4 ,   dice_number=9   )
    _lowerCAmelCase       :   Dict  =    total_frequency_distribution(
        sides_number=6 ,   dice_number=6   )

    _lowerCAmelCase       :   int  =    0
    _lowerCAmelCase       :   Optional[int]  =    9
    _lowerCAmelCase       :   Optional[Any]  =    4 * 9
    _lowerCAmelCase       :   Dict  =    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       :   Tuple  =    (4**9) * (6**6)
    _lowerCAmelCase       :   Union[str, Any]  =    peter_wins_count / total_games_number

    _lowerCAmelCase       :   Any  =    round(_lowerCamelCase ,   ndigits=7   )

    return rounded_peter_win_probability


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








"""simple docstring"""






from collections.abc import Callable







class 	__UpperCamelCase    :

  def __init__(		self    ,_A = None      ):
      '''simple docstring'''

      _lowerCAmelCase       :   list  =    []
      # Stores indexes of each item for supporting updates and deletion.
      _lowerCAmelCase       :   dict  =    {}
      # Stores current size of heap.
      _lowerCAmelCase       :   Union[str, Any]  =    0
      # Stores function used to evaluate the score of an item on which basis ordering
      # will be done.
      _lowerCAmelCase       :   Union[str, Any]  =    key or (lambda _A      : x)

  def __lowerCamelCase		(		self    ,_A      ):
      '''simple docstring'''

      return int((i - 1) / 2      ) if i > 0 else None

  def __lowerCamelCase		(		self    ,_A      ):
      '''simple docstring'''

      _lowerCAmelCase       :   Union[str, Any]  =    int(2 * i + 1      )
      return left if 0 < left < self.size else None

  def __lowerCamelCase		(		self    ,_A      ):
      '''simple docstring'''

      _lowerCAmelCase       :   str  =    int(2 * i + 2      )
      return right if 0 < right < self.size else None

  def __lowerCamelCase		(		self    ,_A    ,_A      ):
      '''simple docstring'''

      _lowerCAmelCase,					_lowerCAmelCase       :   Tuple  =    (
          self.pos_map[self.arr[j][0]],
          self.pos_map[self.arr[i][0]],
      )
      # Then swap the items in the list.
      _lowerCAmelCase,					_lowerCAmelCase       :   Tuple  =    self.arr[j], self.arr[i]

  def __lowerCamelCase		(		self    ,_A    ,_A      ):
      '''simple docstring'''

      return self.arr[i][1] < self.arr[j][1]

  def __lowerCamelCase		(		self    ,_A      ):
      '''simple docstring'''

      _lowerCAmelCase       :   Tuple  =    self._left(_A      )
      _lowerCAmelCase       :   str  =    self._right(_A      )
      _lowerCAmelCase       :   Tuple  =    i

      if left is not None and not self._cmp(_A    ,_A      ):
          _lowerCAmelCase       :   int  =    left
      if right is not None and not self._cmp(_A    ,_A      ):
          _lowerCAmelCase       :   Optional[int]  =    right

      return valid_parent

  def __lowerCamelCase		(		self    ,_A      ):
      '''simple docstring'''

      _lowerCAmelCase       :   Any  =    self._parent(_A      )
      while parent is not None and not self._cmp(_A    ,_A      ):
          self._swap(_A    ,_A      )
          _lowerCAmelCase,					_lowerCAmelCase       :   List[str]  =    parent, self._parent(_A      )

  def __lowerCamelCase		(		self    ,_A      ):
      '''simple docstring'''

      _lowerCAmelCase       :   Union[str, Any]  =    self._get_valid_parent(_A      )
      while valid_parent != index:
          self._swap(_A    ,_A      )
          _lowerCAmelCase,					_lowerCAmelCase       :   Optional[int]  =    valid_parent, self._get_valid_parent(_A      )

  def __lowerCamelCase		(		self    ,_A    ,_A      ):
      '''simple docstring'''

      if item not in self.pos_map:
          return
      _lowerCAmelCase       :   int  =    self.pos_map[item]
      _lowerCAmelCase       :   Dict  =    [item, self.key(_A      )]
      # Make sure heap is right in both up and down direction.
      # Ideally only one of them will make any change.
      self._heapify_up(_A      )
      self._heapify_down(_A      )

  def __lowerCamelCase		(		self    ,_A      ):
      '''simple docstring'''

      if item not in self.pos_map:
          return
      _lowerCAmelCase       :   List[str]  =    self.pos_map[item]
      del self.pos_map[item]
      _lowerCAmelCase       :   Dict  =    self.arr[self.size - 1]
      _lowerCAmelCase       :   Optional[Any]  =    index
      self.size -= 1
      # Make sure heap is right in both up and down direction. Ideally only one
      # of them will make any change- so no performance loss in calling both.
      if self.size > index:
          self._heapify_up(_A      )
          self._heapify_down(_A      )

  def __lowerCamelCase		(		self    ,_A    ,_A      ):
      '''simple docstring'''

      _lowerCAmelCase       :   Tuple  =    len(self.arr      )
      if arr_len == self.size:
          self.arr.append([item, self.key(_A      )]      )
      else:
          _lowerCAmelCase       :   Any  =    [item, self.key(_A      )]
      _lowerCAmelCase       :   str  =    self.size
      self.size += 1
      self._heapify_up(self.size - 1      )

  def __lowerCamelCase		(		self      ):
      '''simple docstring'''

      return self.arr[0] if self.size else None







  def __lowerCamelCase		(		self      ):
      '''simple docstring'''

      _lowerCAmelCase       :   int  =    self.get_top()
      if top_item_tuple:
          self.delete_item(top_item_tuple[0]      )
      return top_item_tuple





def 		lowerCamelCase__		(					):

    '''simple docstring'''





if __name__ == "__main__":
 import doctest

 doctest.testmod()





