

import itertools
import math







def 							_lowerCamelCase   (			SCREAMING_SNAKE_CASE   ):
      '''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 							_lowerCamelCase   (			):
      '''simple docstring'''







      A_							       =		2
      while True:
            if is_prime(SCREAMING_SNAKE_CASE   ):
                  yield num
            num += 1







def 							_lowerCamelCase   (			SCREAMING_SNAKE_CASE = 10001   ):
      '''simple docstring'''







      return next(itertools.islice(prime_generator()		,				nth - 1		,				SCREAMING_SNAKE_CASE   )   )


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








from itertools import product







def 							_lowerCamelCase   (			SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE   ):
      '''simple docstring'''







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

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

      return totals_frequencies







def 							_lowerCamelCase   (			):
      '''simple docstring'''







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

      A_							       =		0
      A_							       =		9
      A_							       =		4 * 9
      A_							       =		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]   )

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

      A_							       =		round(SCREAMING_SNAKE_CASE		,				ndigits=7   )

      return rounded_peter_win_probability


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





