

'''simple docstring'''
import sys
import turtle


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




     return (pa[0] + pa[0]) / 2, (pa[1] + pa[1]) / 2


def 	_lowerCAmelCase (						_UpperCamelCase						:					tuple[float, float]							,			_UpperCamelCase						:					tuple[float, float]							,			_UpperCamelCase						:					tuple[float, float]							,			_UpperCamelCase						:					int							,			)				->      None:
     """simple docstring"""




     my_pen.up()
     my_pen.goto(vertexa[0]							,			vertexa[1]    )
     my_pen.down()
     my_pen.goto(vertexa[0]							,			vertexa[1]    )
     my_pen.goto(vertexa[0]							,			vertexa[1]    )
     my_pen.goto(vertexa[0]							,			vertexa[1]    )

     if depth == 0:
          return

     triangle(_UpperCAmelCase							,			get_mid(_UpperCAmelCase							,			_UpperCAmelCase    )							,			get_mid(_UpperCAmelCase							,			_UpperCAmelCase    )							,			depth - 1    )
     triangle(_UpperCAmelCase							,			get_mid(_UpperCAmelCase							,			_UpperCAmelCase    )							,			get_mid(_UpperCAmelCase							,			_UpperCAmelCase    )							,			depth - 1    )
     triangle(_UpperCAmelCase							,			get_mid(_UpperCAmelCase							,			_UpperCAmelCase    )							,			get_mid(_UpperCAmelCase							,			_UpperCAmelCase    )							,			depth - 1    )


if __name__ == "__main__":
     if len(sys.argv) != 2:
          raise ValueError(
              "Correct format for using this script: "
              "python fractals.py <int:depth_for_fractal>"
          )
     lowerCamelCase			:				str 			=					turtle.Turtle()
     my_pen.ht()
     my_pen.speed(5)
     my_pen.pencolor("red")

     lowerCamelCase			:				Dict 			=					[(-1_7_5, -1_2_5), (0, 1_7_5), (1_7_5, -1_2_5)]  # vertices of triangle
     triangle(vertices[0], vertices[1], vertices[2], int(sys.argv[1]))



'''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))
