perfect_games.py

#!/usr/bin/env python3
# Answer for the 538 Ridder Express for 5/28/21
#
# https://fivethirtyeight.com/features/can-you-crack-the-case-of-the-crystal-key/

N_TEAMS = 30
N_GAMES = 162
EPSILON = 10e-16


def expected_perfect_games(on_base_ptc):
    # Find the odds that any one team would throw a perfect game
    # Note: Given the constraints, in theory both teams could throw a perfect
    # game at the same time, though that is unlikely
    odds_perfect_game = (1 - on_base_ptc) ** 27

    # The expected value of a binomial distribution is p * N, where
    # p is the probability of any given game being perfect, and
    # n is the total number of games thrown
    return (N_TEAMS * N_GAMES) * odds_perfect_game


# Do a binary search for the correct on base percentage
min_obp = 0.0
max_obp = 1.0

while max_obp - min_obp > EPSILON:
    test = (min_obp + max_obp) / 2
    expected_value = expected_perfect_games(test)
    if expected_value > 1:
        min_obp = test
    else:
        max_obp = test

answer = (min_obp + max_obp) / 2
print(f"Result: {answer} ({expected_perfect_games(answer)})")