# Cree par nyckie, le 05/05/2019 en Python 3.4

from math import*
from random import*
def echvar(V,P,n,N):
  A=paramvar(V,P)
  mu=A[0]; sigma=A[2]
  c=0
  for k in range(N):
    m=moysimvar(V,P,n)
    if abs(m-mu)<=2*sigma/sqrt(n):
      c=c+1
  return(c/N)


from math import*
def paramvar(V,P):
  mu=0;Var=0
  for i in range(len(V)):
     mu=mu+V[i]*P[i]
  for i in range(len(V)):
    Var=Var+(V[i]-mu)**2*P[i]
  sigma=sqrt(Var)
  return [mu,Var,sigma]

def simvar(V,P,n):
  N=[0,0,0,0]
  y=P[0]
  z=y+P[1]
  t=z+P[2]
  for i in range(n):
    a=random()
    if 0<a<y:
      N[0]=N[0]+1
    if y<a<z:
      N[1]=N[1]+1
    if z<a<t:
      N[2]=N[2]+1
    if a>t:
      N[3]=N[3]+1
  F=[N[0]/n,N[1]/n,N[2]/n,N[3]/n]
  return(F)


def moysimvar(V,P,n):
  E=0
  F=simvar(V,P,n)
  for i in range(4):
    E=E+V[i]*F[i]
  return E