from math import factorial,exp,sqrt

def permutation(n,r):
  """returns nPr"""
  return factorial(n)/factorial(n-r)

def combination(n,r):
  """Returns nCr"""
  return factorial(n)/(factorial(r)*factorial(n-r))

def binomial_distribution(x,n,p):
  """
  Computes the probability of having x successes out of n trials.
  Each trial has a probability of success p
  """
  nCx = combination(n,x)
  q = 1-p
  return nCx*(p**x)*(q**(n-x))

def geometric_distribution(n,p):
  """
  Computes the probability of 1st success on the nth trial.
  Each trial has a probability of success p
  """
  q = 1 - p
  return p*q**(n-1)

def poisson(k,l):
  """
  Computes the probability of k successes.
  The average number of successes is l
  """
  return (l**k)*exp(-l)/factorial(k)

def erf(x):
  t = 1 / (1 + 0.5*abs(x))
  tau = t*exp( -x**2 - 1.26551223 + 1.00002368*t + 0.37409196*t**2 + 0.09678418*t**3
              - 0.18628806*t**4 + 0.27886807*t**5 -1.13520398*t**6 + 1.48851587*t**7
              - 0.82215223*t**8 + 0.17087277*t**9)
  if x < 0:
    return tau - 1
  else:
    return 1 - tau

def norm(x):
  """
  Computes the probability of X <= x in the normal
  distribution.
  """
  return 0.5*(1 + erf(x/sqrt(2)))