





from itertools import product



def        __lowercase       (				_SCREAMING_SNAKE_CASE			,						_SCREAMING_SNAKE_CASE     )			->       list[int]:


      '''simple docstring'''


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

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

      return totals_frequencies



def        __lowercase       (				)			->       float:


      '''simple docstring'''


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

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

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

      SCREAMING_SNAKE_CASE		=						round(_SCREAMING_SNAKE_CASE			,						ndigits=7     )

      return rounded_peter_win_probability


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









from __future__ import annotations

import math



def        __lowercase       (				_SCREAMING_SNAKE_CASE     )			->       bool:


      '''simple docstring'''


      if 1 < number < 4:
            # 2 and 3 are primes
            return True
      elif number < 2 or number % 2 == 0 or number % 3 == 0:
            # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
            return False

      # All primes number are in format of 6k +/- 1
      for i in range(5			,						int(math.sqrt(_SCREAMING_SNAKE_CASE     ) + 1     )			,						6     ):
            if number % i == 0 or number % (i + 2) == 0:
                  return False
      return True



def        __lowercase       (				_SCREAMING_SNAKE_CASE     )			->       list[int]:


      '''simple docstring'''


      SCREAMING_SNAKE_CASE		=						str(_SCREAMING_SNAKE_CASE     )
      SCREAMING_SNAKE_CASE		=						[n]
      for i in range(1			,						len(_SCREAMING_SNAKE_CASE     )     ):
            list_nums.append(int(str_num[i:]     )     )
            list_nums.append(int(str_num[:-i]     )     )
      return list_nums



def        __lowercase       (				_SCREAMING_SNAKE_CASE     )			->       bool:


      '''simple docstring'''


      if len(str(_SCREAMING_SNAKE_CASE     )     ) > 3:
            if not is_prime(int(str(_SCREAMING_SNAKE_CASE     )[-3:]     )     ) or not is_prime(int(str(_SCREAMING_SNAKE_CASE     )[:3]     )     ):
                  return False
      return True



def        __lowercase       (				_SCREAMING_SNAKE_CASE = 11     )			->       list[int]:


      '''simple docstring'''


      SCREAMING_SNAKE_CASE		=						[]
      SCREAMING_SNAKE_CASE		=						13
      while len(_SCREAMING_SNAKE_CASE     ) != count:
            if validate(_SCREAMING_SNAKE_CASE     ):
                  SCREAMING_SNAKE_CASE		=						list_truncated_nums(_SCREAMING_SNAKE_CASE     )
                  if all(is_prime(_SCREAMING_SNAKE_CASE     ) for i in list_nums     ):
                        list_truncated_primes.append(_SCREAMING_SNAKE_CASE     )
            num += 2
      return list_truncated_primes



def        __lowercase       (				)			->       int:


      '''simple docstring'''


      return sum(compute_truncated_primes(11     )     )


if __name__ == "__main__":
      print(F'''{sum(compute_truncated_primes(1_1)) = }''')


