
from itertools import product


def       A       (					lowercase    ,       lowercase )   ->  list[int]:
		'''simple docstring'''

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

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

		return totals_frequencies


def       A       (					)   ->  float:
		'''simple docstring'''

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

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

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

		UpperCamelCase		=     round(lowercase    ,       ndigits=7 )

		return rounded_peter_win_probability


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


from random import randint
from tempfile import TemporaryFile

import numpy as np


def       A       (					lowercase    ,       lowercase    ,       lowercase )   ->  str:
		'''simple docstring'''

		UpperCamelCase		=     0
		if start < end:
				UpperCamelCase		=     randint(lowercase    ,       lowercase )
				UpperCamelCase		=     a[end]
				UpperCamelCase		=     a[pivot]
				UpperCamelCase		=     temp

				UpperCamelCase	,  UpperCamelCase		=     _in_place_partition(lowercase    ,       lowercase    ,       lowercase )
				count += _in_place_quick_sort(lowercase    ,       lowercase    ,       p - 1 )
				count += _in_place_quick_sort(lowercase    ,       p + 1    ,       lowercase )
		return count


def       A       (					lowercase    ,       lowercase    ,       lowercase )   ->  int:
		'''simple docstring'''

		UpperCamelCase		=     0
		UpperCamelCase		=     randint(lowercase    ,       lowercase )
		UpperCamelCase		=     a[end]
		UpperCamelCase		=     a[pivot]
		UpperCamelCase		=     temp
		UpperCamelCase		=     start - 1
		for index in range(lowercase    ,       lowercase ):
				count += 1
				if a[index] < a[end]:  # check if current val is less than pivot value
						UpperCamelCase		=     new_pivot_index + 1
						UpperCamelCase		=     a[new_pivot_index]
						UpperCamelCase		=     a[index]
						UpperCamelCase		=     temp

		UpperCamelCase		=     a[new_pivot_index + 1]
		UpperCamelCase		=     a[end]
		UpperCamelCase		=     temp
		return new_pivot_index + 1, count


_UpperCAmelCase					: Union[str, Any]      =       TemporaryFile()
_UpperCAmelCase					: List[Any]      =       100  # 1000 elements are to be sorted


_UpperCAmelCase   ,_UpperCAmelCase					: Any      =       0, 1  # mean and standard deviation
_UpperCAmelCase					: Any      =       np.random.normal(mu, sigma, p)
np.save(outfile, X)
print("The array is")
print(X)


outfile.seek(0)  # using the same array
_UpperCAmelCase					: Any      =       np.load(outfile)
_UpperCAmelCase					: str      =       len(M) - 1
_UpperCAmelCase					: List[str]      =       _in_place_quick_sort(M, 0, r)

print(
    "No of Comparisons for 100 elements selected from a standard normal distribution"
    "is :"
)


print(z)
