Probabilistic Finality: The Math of the 6-Confirmation Rule

Unlike traditional banking systems where a transaction is either "settled" or "not," Bitcoin utilizes Probabilistic Finality. As a transaction is buried under more blocks (increasing block height), the probability of that transaction being reversed via a "Most Work" chain reorganization drops exponentially.

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📉 1. The Attacker's Race

In Section 11 of the Bitcoin Whitepaper, Satoshi modeled the probability of an attacker catching up from behind.

Imagine an attacker who has a percentage $q$ of the total hashrate, while the honest network has $p$. If the attacker is $z$ blocks behind, the probability $P$ that they can ever catch up and override the main chain is:

$$P = \begin{cases} 1 & \text{if } p \le q \ (q/p)^z & \text{if } p > q \end{cases}$$

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🎲 2. The Poisson Distribution Model

The discovery of blocks is a Poisson Process. Even if an attacker has less hashrate than the honest network, they might get "lucky" and find several blocks in a row.

To calculate the risk, we must account for the attacker's expected number of blocks vs. the number of blocks the honest network has already moved forward.

Confirmations ($z$)	Attacker Power ($q=10\%$)	Attacker Power ($q=30\%$)
1	0.2045	0.5311
3	0.0248	0.2464
6	0.0006	0.0766
10	0.0000	0.0135

As shown, if an attacker has $10\%$ of the network's hashrate, the chance of them reversing a transaction with 6 confirmations is less than 0.1%.

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🛡️ 3. Why 6 Confirmations?

The "6 blocks" standard (approx. 1 hour) was chosen as a conservative threshold for medium-to-high value transactions. At this depth, even an attacker with substantial hashrate (e.g., $25\%$) would need an incredible streak of luck to overtake the honest chain.

For micro-payments (like buying a coffee), merchants often accept 0-conf (zero confirmations) or 1-conf, accepting the tiny risk of a double-spend in exchange for speed.

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🕰️ 4. The Concept of "Deep" Reorgs

A reorg of 1 or 2 blocks is relatively common due to natural network latency (orphans). A reorg of 6+ blocks is extremely rare and usually indicates a massive network failure, a significant 51% attack, or a major protocol bug.

This is why, after 100 blocks (the maturity rule), the block reward becomes spendable, as the "finality" of those coins is considered mathematically absolute for all practical purposes.