

'''simple docstring'''
from __future__ import annotations


def 	_lowerCAmelCase (						_UpperCamelCase						:					int							,			_UpperCamelCase						:					int    )				->      tuple[int, int]:
     """simple docstring"""




     if b == 0:
          return (1, 0)
     ((_SCREAMING_SNAKE_CASE)   ,						(_SCREAMING_SNAKE_CASE))							   =extended_euclid(_UpperCamelCase							,			a % b    )
     _SCREAMING_SNAKE_CASE							   =a // b
     return (y, x - k * y)


def 	_lowerCAmelCase (						_UpperCamelCase						:					int							,			_UpperCamelCase						:					int							,			_UpperCamelCase						:					int							,			_UpperCamelCase						:					int    )				->      int:
     """simple docstring"""




     ((_SCREAMING_SNAKE_CASE)   ,						(_SCREAMING_SNAKE_CASE))							   =extended_euclid(_UpperCamelCase							,			_UpperCamelCase    )
     _SCREAMING_SNAKE_CASE							   =na * na
     _SCREAMING_SNAKE_CASE							   =ra * x * na + ra * y * na
     return (n % m + m) % m


def 	_lowerCAmelCase (						_UpperCamelCase						:					int							,			_UpperCamelCase						:					int    )				->      int:
     """simple docstring"""




     ((_SCREAMING_SNAKE_CASE)   ,						(_SCREAMING_SNAKE_CASE))							   =extended_euclid(_UpperCamelCase							,			_UpperCamelCase    )
     if b < 0:
          _SCREAMING_SNAKE_CASE							   =(b % n + n) % n
     return b


def 	_lowerCAmelCase (						_UpperCamelCase						:					int							,			_UpperCamelCase						:					int							,			_UpperCamelCase						:					int							,			_UpperCamelCase						:					int    )				->      int:
     """simple docstring"""




     _SCREAMING_SNAKE_CASE   ,						_SCREAMING_SNAKE_CASE							   =invert_modulo(_UpperCamelCase							,			_UpperCamelCase    ), invert_modulo(_UpperCamelCase							,			_UpperCamelCase    )
     _SCREAMING_SNAKE_CASE							   =na * na
     _SCREAMING_SNAKE_CASE							   =ra * x * na + ra * y * na
     return (n % m + m) % m


if __name__ == "__main__":
     from doctest import testmod

     testmod(name="chinese_remainder_theorem", verbose=True)
     testmod(name="chinese_remainder_theorem2", verbose=True)
     testmod(name="invert_modulo", verbose=True)
     testmod(name="extended_euclid", verbose=True)



'''simple docstring'''
from math import factorial


def 	_lowerCAmelCase (						_UpperCamelCase						:					int							,			_UpperCamelCase						:					int							,			_UpperCamelCase						:					float    )				->      float:
     """simple docstring"""




     if successes > trials:
          raise ValueError('successes must be lower or equal to trials'    )
     if trials < 0 or successes < 0:
          raise ValueError('the function is defined for non-negative integers'    )
     if not isinstance(_UpperCamelCase							,			_UpperCamelCase    ) or not isinstance(_UpperCamelCase							,			_UpperCamelCase    ):
          raise ValueError('the function is defined for non-negative integers'    )
     if not 0 < prob < 1:
          raise ValueError('prob has to be in range of 1 - 0'    )
     _SCREAMING_SNAKE_CASE							   =(prob**successes) * ((1 - prob) ** (trials - successes))
     # Calculate the binomial coefficient: n! / k!(n-k)!
     _SCREAMING_SNAKE_CASE							   =float(factorial(_UpperCamelCase    )    )
     coefficient /= factorial(_UpperCamelCase    ) * factorial(trials - successes    )
     return probability * coefficient


if __name__ == "__main__":
     from doctest import testmod

     testmod()
     print("Probability of 2 successes out of 4 trails")
     print("with probability of 0.75 is:", end=" ")
     print(binomial_distribution(2, 4, 0.7_5))
