prob.py

from itertools import product
best = 0
ta = 0
tb = 0
for a,b in product(range(0,11), range(0,11)):
    if (a == 0 and b == 0) or (a == 10 and b == 10):
        rate = 0.5
    else:
        rate = 0.5 * (a/(a+b) + (10-a)/(20-a-b))
    if rate > best:
        ta, tb = a, b
        best = rate

    print("a: {}, b: {}, rate: {}".format(a,b,rate))

print("best is: {},  with a,b = {}, {}".format(best, ta, tb))