An edge is a small number

def edge(p, win, loss):
    return p * win - (1 - p) * loss

rows = [
    ("52% coin, even money", 0.52, 1.0, 1.0),
    ("40% win rate, 2:1 payoff", 0.40, 2.0, 1.0),
    ("60% win rate, 1:2 payoff", 0.60, 0.5, 1.0),
    ("55% coin, even money", 0.55, 1.0, 1.0),
]
for name, p, w, l in rows:
    e = edge(p, w, l)
    print(f"{name:26s}: edge {e:+.3f} per $1 risked -> {'profitable' if e > 0 else 'LOSING'}")

print(f"breakeven win rate at 2:1 payoff: {1 / (1 + 2):.1%}")

import numpy as np

rng = np.random.default_rng(7)
edge_coin = np.where(rng.random(2500) < 0.52, 1, -1)
fair_a = np.where(rng.random(2500) < 0.50, 1, -1)
fair_b = np.where(rng.random(2500) < 0.50, 1, -1)

for name, coin in [("52% coin", edge_coin), ("fair A", fair_a), ("fair B", fair_b)]:
    print(f"{name:8s}: final PnL {coin.sum():+5d}, win rate {(coin > 0).mean():.4f}")

# how often is the 52% coin DOWN, across alternate histories?
rng = np.random.default_rng(3)
after_100 = np.where(rng.random((100_000, 100)) < 0.52, 1, -1).sum(axis=1)
print(f"P(down after   100 flips): {(after_100 < 0).mean():.1%}")
after_2401 = np.where(rng.random((20_000, 2401)) < 0.52, 1, -1).sum(axis=1)
print(f"P(down after 2,401 flips): {(after_2401 < 0).mean():.1%}")

import numpy as np

rng = np.random.default_rng(3)
coins = np.where(rng.random((100, 250)) < 0.50, 1, -1)   # 100 FAIR coins
win_rates = (coins > 0).mean(axis=1)

print("true win rate of every coin: 50.0%")
print(f"best of 100 after 250 flips: {win_rates.max():.1%}")
print(f"worst                      : {win_rates.min():.1%}")
print(f"coins above 52%            : {(win_rates > 0.52).sum()} of 100")