

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

from math import gcd


def 	_lowerCAmelCase (						_UpperCamelCase						:					int							,			_UpperCamelCase						:					int = 2							,			_UpperCamelCase						:					int = 1							,			_UpperCamelCase						:					int = 3							,			)				->      int | None:
     """simple docstring"""




     if num < 2:
          raise ValueError('The input value cannot be less than 2'    )

     # Because of the relationship between ``f(f(x))`` and ``f(x)``, this
     # algorithm struggles to find factors that are divisible by two.
     # As a workaround, we specifically check for two and even inputs.
     #   See: https://math.stackexchange.com/a/2856214/165820
     if num > 2 and num % 2 == 0:
          return 2

     # Pollard's Rho algorithm requires a function that returns pseudorandom
     # values between 0 <= X < ``num``.  It doesn't need to be random in the
     # sense that the output value is cryptographically secure or difficult
     # to calculate, it only needs to be random in the sense that all output
     # values should be equally likely to appear.
     # For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
     # However, the success of Pollard's algorithm isn't guaranteed and is
     # determined in part by the initial seed and the chosen random function.
     # To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
     # where ``C`` is a value that we can modify between each attempt.
     def rand_fn(_UpperCamelCase						:					int							,			_UpperCamelCase						:					int							,			_UpperCamelCase						:					int    ) -> int:
          return (pow(_UpperCamelCase							,			2    ) + step) % modulus

     for _ in range(_UpperCamelCase    ):
          # These track the position within the cycle detection logic.
          _SCREAMING_SNAKE_CASE							   =seed
          _SCREAMING_SNAKE_CASE							   =seed

          while True:
               # At each iteration, the tortoise moves one step and the hare moves two.
               _SCREAMING_SNAKE_CASE							   =rand_fn(_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase    )
               _SCREAMING_SNAKE_CASE							   =rand_fn(_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase    )
               _SCREAMING_SNAKE_CASE							   =rand_fn(_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase    )

               # At some point both the tortoise and the hare will enter a cycle whose
               # length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
               # the tortoise and hare will end up on the same value modulo ``p``.
               # We can detect when this happens because the position difference between
               # the tortoise and the hare will share a common divisor with ``num``.
               _SCREAMING_SNAKE_CASE							   =gcd(hare - tortoise							,			_UpperCamelCase    )

               if divisor == 1:
                    # No common divisor yet, just keep searching.
                    continue
               else:
                    # We found a common divisor!
                    if divisor == num:
                         # Unfortunately, the divisor is ``num`` itself and is useless.
                         break
                    else:
                         # The divisor is a nontrivial factor of ``num``!
                         return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
          _SCREAMING_SNAKE_CASE							   =hare

          # The new step value for the random function can just be incremented.
          # At first the results will be similar to what the old function would
          # have produced, but the value will quickly diverge after a bit.
          step += 1

     # We haven't found a divisor within the requested number of attempts.
     # We were unlucky or ``num`` itself is actually prime.
     return None


if __name__ == "__main__":
     import argparse

     lowerCamelCase			:				str 			=					argparse.ArgumentParser()
     parser.add_argument(
         "num",
         type=int,
         help="The value to find a divisor of",
     )
     parser.add_argument(
         "--attempts",
         type=int,
         default=3,
         help="The number of attempts before giving up",
     )
     lowerCamelCase			:				Union[str, Any] 			=					parser.parse_args()

     lowerCamelCase			:				Dict 			=					pollard_rho(args.num, attempts=args.attempts)
     if divisor is None:
          print(f'''{args.num} is probably prime''')
     else:
          lowerCamelCase			:				int 			=					args.num // divisor
          print(f'''{args.num} = {divisor} * {quotient}''')



'''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))
