


from math import factorial
def 							_UpperCAmelCase				(UpperCamelCase__     :  int				,							UpperCamelCase__     :  int				,							UpperCamelCase__     :  float						):
       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"						)
       _A      :       Optional[Any]				    = (prob**successes) * ((1 - prob) ** (trials - successes))
       # Calculate the binomial coefficient: n! / k!(n-k)!
       _A      :       Dict				    = 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, 0.75))





'''simple docstring'''






from __future__ import annotations

import matplotlib.pyplot as plt  # type: ignore
import numpy

# initial triangle of Koch snowflake
UpperCamelCase__:   Tuple								=			numpy.array([0, 0])
UpperCamelCase__:   Union[str, Any]								=			numpy.array([0.5, 0.8660254])
UpperCamelCase__:   Dict								=			numpy.array([1, 0])
UpperCamelCase__:   int								=			[VECTOR_1, VECTOR_2, VECTOR_3, VECTOR_1]


def  snake_case_     (					_lowerCAmelCase  :						list[numpy.ndarray]   ,      _lowerCAmelCase  :						int  )      ->   list[numpy.ndarray]:
							UpperCAmelCase :       Union[str, Any] 							=		initial_vectors
							for _ in range(_lowerCAmelCase  ):
														UpperCAmelCase :       Union[str, Any] 							=		iteration_step(_lowerCAmelCase  )
							return vectors


def  snake_case_     (					_lowerCAmelCase  :						list[numpy.ndarray]  )      ->   list[numpy.ndarray]:
							UpperCAmelCase :       Tuple 							=		[]
							for i, start_vector in enumerate(vectors[:-1]  ):
														UpperCAmelCase :       List[str] 							=		vectors[i + 1]
														new_vectors.append(_lowerCAmelCase  )
														UpperCAmelCase :       Optional[Any] 							=		end_vector - start_vector
														new_vectors.append(start_vector + difference_vector / 3  )
														new_vectors.append(
														    start_vector + difference_vector / 3 + rotate(difference_vector / 3   ,      60  )  )
														new_vectors.append(start_vector + difference_vector * 2 / 3  )
							new_vectors.append(vectors[-1]  )
							return new_vectors


def  snake_case_     (					_lowerCAmelCase  :						numpy.ndarray   ,      _lowerCAmelCase  :						float  )      ->   numpy.ndarray:
							UpperCAmelCase :       List[str] 							=		numpy.radians(_lowerCAmelCase  )
							UpperCAmelCase							,		UpperCAmelCase :       Tuple 							=		numpy.cos(_lowerCAmelCase  ), numpy.sin(_lowerCAmelCase  )
							UpperCAmelCase :       Union[str, Any] 							=		numpy.array(((c, -s), (s, c))  )
							return numpy.dot(_lowerCAmelCase   ,      _lowerCAmelCase  )


def  snake_case_     (					_lowerCAmelCase  :						list[numpy.ndarray]  )      ->   None:
							UpperCAmelCase :       List[Any] 							=		plt.gca()
							axes.set_aspect('''equal'''  )

							# matplotlib.pyplot.plot takes a list of all x-coordinates and a list of all
							# y-coordinates as inputs, which are constructed from the vector-list using
							# zip()
							UpperCAmelCase							,		UpperCAmelCase :       str 							=		zip(*_lowerCAmelCase  )
							plt.plot(_lowerCAmelCase   ,      _lowerCAmelCase  )
							plt.show()


if __name__ == "__main__":
						import doctest

						doctest.testmod()

						UpperCamelCase__:   List[Any]								=			iterate(INITIAL_VECTORS, 5)
						plot(processed_vectors)





