Confirmations and Finality

# Probability that an attacker with q < 0.5 fraction of hashrate
# successfully replaces N confirmations (Nakamoto's whitepaper formula):

import math

def reorg_probability(q: float, n: int) -> float:
    """Probability attacker controlling fraction q (< 0.5) of hashrate
    replaces n confirmations. From Bitcoin whitepaper."""
    p = 1 - q
    lam = n * (q / p)
    total = 0.0
    for k in range(n + 1):
        # Poisson PMF
        pk = (lam ** k) * math.exp(-lam) / math.factorial(k)
        # Conditional probability of catching up
        if k >= n:
            qk = 1.0
        else:
            qk = (q / p) ** (n - k)
        total += pk * qk
    return total

# Example: 10% attacker
for n in [1, 2, 6, 10]:
    p = reorg_probability(0.10, n)
    print(f"q=10%, n={n}: P(success) = {p:.6f}")

# Output (approx):
# q=10%, n=1: P(success) = 0.205
# q=10%, n=2: P(success) = 0.051
# q=10%, n=6: P(success) = 0.00059
# q=10%, n=10: P(success) = 0.0000122

# For 30% attacker, even 10 confirms is risky (~5% reorg probability).
# For 1% attacker, 1 confirm is fine for small amounts.
# Standard exchange rule: 6 confirms (about 1 hour) for safety against
# any realistic adversary.