Because the teams have random, uniform, and independent winning chances, and because they lie on a continuous distribution, there is a strict order of winning percentages for all the teams. Thus, we can treat this as a combinatorics problem, where teams from the same division are treated as indistinguishable.
If we let C represent an Enigma League Central division team and let E represent an Enigma League East division team, we can find the probability that CCCCCEEEEE arises from all possible permutations of said letters, as this is the only configuration where all East teams have higher winning percentages than all Central teams.
As there are a total of
Answer: 1/252
When my rumination yields no destination, it's a simulation that is my salvation.
import random
from collections import defaultdict
N = 10**7
total = 0
NUM_DIVS = 6
NUM_TEAMS_PER_DIV = 5
for t in range(N):
if t > 0 and t % 10**5 == 0:
print(total, t, total / t)
seen = defaultdict(lambda: 0)
for div in random.sample(
list(range(NUM_DIVS)) * NUM_TEAMS_PER_DIV, NUM_DIVS * NUM_TEAMS_PER_DIV
):
seen[div] += 1
if seen[div] == NUM_TEAMS_PER_DIV:
total += len(seen) != NUM_DIVS
break
print(total, N, total / N)Answer: 0.0862