How to Calculate the Probability of a Successful 51% Attack

A 51% attack (or majority attack) represents the most fundamental security threat to a Proof-of-Work (PoW) or Proof-of-Stake (PoS) blockchain. It occurs when a single entity or colluding group gains control of more than half (51%) of the network's total hashing power (in PoW) or staked cryptocurrency (in PoS). This majority control allows the attacker to manipulate the block finalization process, most notably to perform a double-spending attack by reversing transactions that have already been confirmed on the honest chain

Calculating the probability of a successful 51% attack is complex because it is less about a static probability of gaining 51% control and more about the probability of winning a block-race after control is achieved. For PoW, this probability is modeled using discrete-time random walk or the Gambler's Ruin Problem, which measures the attacker's likelihood of privately mining a longer chain than the honest network before their fraudulent transaction is confirmed by too many blocks. The primary factors influencing this probability are the attacker's relative mining power and the number of blocks the victim waits for confirmation.

How to Calculate the Probability of a Successful 51% Attack

Probability Model for PoW (Hash Power)

In a Proof-of-Work (PoW) system, the success probability for a double-spending attack is primarily modeled by the attacker's relative hash rate (\alpha) and the number of confirmation blocks (z) the victim waits for. The probability (q) that the attacker finds the next block is \alpha, and the probability (p) that the honest network finds the next block is 1 - \alpha. The probability of the attacker successfully catching up to the honest chain (P_{z}) when they are z blocks behind is given by:

This formula demonstrates the core dynamic: if the attacker controls exactly 50% (\alpha = 0.5), the probability q = p, meaning P_z = 1, which suggests a catch-up is guaranteed over infinite time. If the attacker has less than 50% (\alpha < 0.5), the probability of catching up drops exponentially as the number of confirmation blocks (z) increases. Crucially, a successful 51% attack is often defined as having \alpha > 0.5, which makes the probability of the attacker eventually building a longer chain (and successfully double-spending) virtually 100% if they can maintain the majority control long enough. For a mature network like Bitcoin, the cost of acquiring \alpha > 0.5 is the actual security barrier.

Probability Model for PoS (Economic Cost)

For Proof-of-Stake (PoS) systems, the probability calculation shifts from computational power (hash rate) to economic power (staked currency). The success probability is highly dependent on an attacker's ability to acquire and maintain a majority of the total staked coins, \ge 51\%. Unlike PoW where an attacker can rent hash power, a PoS attack requires purchasing or otherwise acquiring the majority of the native asset, making it a direct function of the asset's market capitalization.

The calculation of success is less about a probabilistic block-race and more about the financial feasibility and reversibility of the attack. The probability is considered near-zero for large PoS chains because the cost to acquire 51\% of the staked tokens is prohibitively high—often billions of dollars. Furthermore, most PoS systems include a slashing mechanism, which penalizes (destroys) the attacker's staked coins upon detection of malicious behavior. This means the financial penalty is so severe that the expected value of the attack becomes negative, effectively reducing the probability of a rational actor attempting the attack to zero.

Conclusion

The probability of a successful 51% attack is ultimately a function of cost-to-benefit analysis for a rational attacker. In Proof-of-Work, the probability hinges on the relative hash power (\alpha) and the confirmation depth (z), modeled exponentially to show that the attacker's chances drop rapidly for \alpha < 0.5 as more blocks are added. The immense cost of physical hardware and electricity for major PoW chains serves as the primary defense against \alpha > 0.5.

In contrast, the probability of a successful 51% attack in Proof-of-Stake is primarily an economic calculation. The probability is minimized by the prohibitive cost of acquiring 51\% of the staked tokens (Market Cap \times Staking Ratio) and the powerful deterrent of economic slashing. Therefore, while the mathematical probability exists for both, the real-world probability is a measure of the network's economic security and market depth.

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