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from random import random
from math import sqrt

val =    [  1 ,  2 ,   5,  -3 ]
proba =  [ 0.1, 0.5, 0.3, 0.1 ]

def simul_tirage(val, proba):
  t = random()
  s = proba[0]
  k = 0
  while t > s:
    k = k + 1
    s = s + proba[k]
  return val[k]

def simul_ech(val, proba, n):
  res = [0,0,0,0]
  for J in range (n):
    tirage = simul_tirage(val, proba)
    for k in range(4):
      if val[k] == tirage:
        res[k] = res[k] + 1 
  for k in range(4):
    res[k] = ...
  return res

def esp_var(val, proba):
  s = 0
  for k in range(4):
    s = s + val[k]*proba[k]
  return s

def var_var(val, proba):
  s = 0
  e = esp_var(val, proba)
  for k in range(4):
    ...
  return s

def proportions(val, proba, n, N):
  res = [0,0,0]
  mu = esp_var(val, proba)
  sigma = sqrt(var_var(val, proba))
  for j in range (N):
    ech = simul_ech(val, proba, n)
    moy = esp_var( val, ech)
    if abs(moy - mu) < sigma/sqrt(n):
      res[0] = res[0] + 1
    if ... :
      res[1] = res[1] + 1
    if ... :
      res[2] = res[2] + 1
  for k in range(3):
    res[k] = res[k]/N
  return res

# simulation
print(proportions(val, proba, 100, 1000))