from fractions import Fraction

Bg1 = {"Red": 4, "Black": 5} Bg2 = {"Red": 3, "Black": 7}

Case_1 = Fraction(Bg1["Red"], Bg1["Red"] + Bg1["Black"]) * Fraction(Bg2["Black"], Bg2["Red"] + Bg2["Black"]) * Fraction(Bg2["Black"]-1, Bg2["Red"] + Bg2["Black"]-1)

Case_2 = Fraction(Bg1["Black"], Bg1["Red"] + Bg1["Black"]) * Fraction(Bg2["Black"], Bg2["Red"] + Bg2["Black"]) * Fraction(Bg2["Red"], Bg2["Red"] + Bg2["Black"]-1)

Case_3 = Fraction(Bg1["Black"], Bg1["Red"] + Bg1["Black"]) * Fraction(Bg2["Black"], Bg2["Red"] + Bg2["Black"]-1) * Fraction(Bg2["Red"], Bg2["Red"] + Bg2["Black"])

print(Case_1 + Case_2 + Case_3)

I tried to do it both pythonically and with clear decipherability from the problem description. Pythonically meant, in this context, making good use of the unpacking operators.

from math import gcd
from itertools import product, combinations

b1 = (*'r'*4, *'b'*5)
b2 = (*'r'*3, *'b'*7)
total = [(*a, *b) for a, b in product(combinations(b1, 1), combinations(b2, 2))]
n = len([_ for _ in total if _.count('b') == 2])
d = len(total)
factor = gcd(n, d)
print(n//factor, '/', d//factor, sep='')

from math import comb

total_ways = comb(9, 1) * comb(10, 2)

ways_case_A = comb(4, 1) * comb(7, 2)

ways_case_B = comb(5, 1) * comb(7, 1) * comb(3, 1)

successful_ways = ways_case_A + ways_case_B

from fractions import Fraction probability = Fraction(successful_ways, total_ways)

print(probability)

check out the link for full working code in python with explanation of where it's used in AI and ML, so probability is important while learning AI and ML, https://youtu.be/M3WcDNj4YJ8?si=2xMBBjqkuBsSoo0c

Use binomial coefficient

from math import gcd

def prob(red_1, black_1, red_2, black_2):
    
    # Total outcomes
    total_1 = red_1 + black_1
    total_2 = red_2 + black_2
  
    total_outcomes = total_1 * (total_2 * (total_2 - 1) // 2)
    
    # Favorable outcomes
    # Black from Bag 1, 1 red + 1 black from Bag 2
    BRB = black_1 * (red_2 * black_2)   
    
    # Red from Bag 1, 2 blacks from Bag 2
    RBB = red_1 * (black_2 * (black_2 - 1) // 2)
    
    favorables = BRB + RBB
    
    divisor = gcd(favorables, total_outcomes)
    num = favorables // divisor
    den = total_outcomes // divisor
    
    return f"{num}/{den}"