51% Attack & Double-Spend Race Simulator

Racing…

Attack Parameters

Attacker hash power 30%

Confirmations required (z) 3

Simulation speed 1.0×

Live Race

Honest chain

0 blocks

Attacker chain

0 blocks

Honest lead

Merchant ships?

Not yet

Statistics (Monte Carlo)

Trials run

Attacks won

Empirical success

—

Nakamoto formula

—

Honest network chain

Attacker's private chain

Confirmation threshold (z)

How it works

Both the honest network and the attacker mine blocks as independent Poisson processes: block-finding events arrive randomly over time at a rate proportional to hash power. If the attacker controls a fraction q of total hash power, the honest network controls p = 1 − q, and both race to extend a chain from the same fork point (the moment of the double-spend transaction).

A merchant waits for z confirmations (blocks built on top of the transaction) before releasing goods. The attacker mines secretly and only reveals their chain if it ever becomes at least as long as the honest chain — triggering a reorg that erases the confirmed transaction and lets the attacker double-spend the coins.

Satoshi Nakamoto's original paper gives the exact probability the attacker eventually catches up, using a Poisson distribution with mean λ = z·(q/p):

P(success) = 1 − Σ(k=0..z) [ e^-λ λ^k / k! · (1 − (q/p)^(z-k)) ]
λ = z · q/p            (q < p; if q ≥ p, P = 1)

Attacker hash power — the attacker's share q of total network hash rate. Above 50% the attacker is guaranteed to eventually win regardless of z (this is the "51% attack").
Confirmations (z) — how many blocks the merchant waits for. Each extra confirmation shrinks the attacker's catch-up odds roughly geometrically when q < p.
New Race — restarts a single race from the fork; Auto-Run Trials replays races back-to-back so the empirical win rate converges to the formula above.

Real-world relevance: this is the exact security model behind Bitcoin's "wait for N confirmations" convention — exchanges typically require 3–6 confirmations because with well under 50% attacker hash power, P(success) drops below any practical threshold within a handful of blocks.

honest rate — · attacker rate —

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