from fractions import Fraction

def urns(a):
    prob = [Fraction(x[0], sum(x)) for x in a]
    prob_c = [1 - x for x in prob]
    
    n = len(a)
    ans = 0
    for i in range(n):
        p = prob_c[i] * prob[(i + 1) % n] * prob[(i + 2) % n]
        ans += p
    return ans

b =  [[4, 3], [5, 4], [4, 4]]
print(urns(b))

from fractions import Fraction
urns = [[4,3],[5,4],[4,4]]
red = []
black = []
for lst in urns:
    red.append(Fraction(lst[0],sum(lst)))
    black.append(Fraction(lst[1],sum(lst)))
doub = []
for i in range(len(red)-1):
    for j in range(i+1, len(red)):
        doub.append(red[i]*red[j])
final = []
for j in range(len(black)):
    final.append(doub[j]*black[len(black)-j-1])
print(sum(final))

from itertools import combinations
from fractions import Fraction

n = 3
k = 2

p_r = [Fraction(4,7), Fraction(5,9), Fraction(4,8)]
p_b = [Fraction(3,7), Fraction(4,9), Fraction(4,8)]
total = Fraction(0,1)

for i,j in combinations(range(n), k):
    remaining_option = next(x for x in range(n) if x not in (i, j))
    total += p_r[i] * p_r[j] * p_b[remaining_option]

print(total)

X = ['r','r','r','r','b','b','b']
Y = ['r','r','r','r','r','b','b','b','b']
Z = ['r','r','r','r','b','b','b','b']

matched = 0
instance = 0

for x in X:
    for y in Y:
        for z in Z:
            if ([x,y,z].count('r')==2) and ([x,y,z].count('b')==1):
                matched += 1
            instance += 1

print(str(matched//12)+"/"+str(instance//12))

from fractions import Fraction

p1 = ['r']*4 + ['b']*3
p2 = ['r']*5 + ['b']*4
p3 = ['r']*4 + ['b']*4

combo = []
visited = 0

for i in p1:
    for j in p2:
        for k in p3:
            temp = [i, j, k]
            count_r = temp.count('r')
            count_b = temp.count('b')
            if (count_r==2) and (count_b==1):
                combo.append(temp)
                
            visited += 1

print(Fraction(len(combo), visited))