from sympy.interactive import printing
printing.init_printing()
import sympy as sym
from sympy.mpmath import limit, inf
from sympy import Rational as frac
%pylab inline --no-import-all

1-frac(5,6)**4

1-(5.0/6)**4

1-frac(35,36)**24

1-(35.0/36)**24

def p_a(n):
    denom = 6**n
    num = denom - 1
    return 1 - (num/denom)**(2*denom/3)

x = np.linspace(1,6)
plt.plot(x, p_a(x))
None

limit(p_a, inf)

frac(2,3)*frac(4,52)*frac(48,51)

from itertools import permutations

# Les lettres et les boîtes ont les indices 0, 1, 2, etc.
# Une lettre est dans boîte correcte quand les deux ont
# la même indice.
def au_moins_une_lettre_correcte(lst):
    for ind, val in enumerate(lst):
        if ind == val:
            return True
    return False

# Quand il y a N boîtes, quelle fraction des distributions sont correctes?
def compter_distributions(n):
    distributions = list(permutations(range(0,n)))
    correctes = [d for d in distributions if au_moins_une_lettre_correcte(d)]
    return [len(correctes), len(distributions)]

map(compter_distributions, [1,2,3,4,5,6])

list(permutations(range(0,2)))

list(permutations(range(0,3)))

list(permutations(range(0,4)))

(frac(1,1) - frac(1,2) + frac(1,6) - frac(1,24)) * 24

from sympy import binomial as bn, factorial as fact

def probabilite_au_moins_une_correcte(n):
  somme = 0
  for k in range(1,n+1):
    somme += (-1)**(k+1) * frac(1, fact(k))
  return somme

[fact(i) * probabilite_au_moins_une_correcte(i) for i in range(1,7)]