user105482: I thought hashing power is an expression of the probability of being the one to solve the cryptographic puzzle and placing a new block on the network.
If that's true, owning 51% of the hashing power should just give a slightly higher chance of mining a new block than 49% hashing power. However when I read articles explaining Bitcoin and mining, owning 51% of the hashing power in the network is presented as a guarantee to be able to outpace the rest of the network, to the point where an alternate chain can be created and funds double spent.
Given what I assume (could be wrong) is a marginally higher probability of solving the puzzle first at 51%, how does that make sense?
Couldn't someone who got a little lucky do that with 40% of the hashing power? How does 51% of the hashing power guarantee that one will be able to carry out an attack on the network?

user208: If you do not bound the time the attacker has, if he has 51% of the hash power (and is able to hold that majority during the entire attack time), he will eventually always overtake the rest of the network.
More formally: the probability of the attacker overtaking the honest network will converge to 100% as the attack time goes to infinity. There is no guarantee that this happens quickly at all, and that probability will never reach 100%, but it will come arbitrarily close.
Someone with 51% of the hash rate can simply produce blocks faster than the rest of the network combined, so this should not be a surprise. Sure, he may be unlucky for the first block, or the second, or even the first 100 blocks, but in the long term, the rest of the network simply cannot keep up with the attacker.

user63872: With any probability distribution, a larger sample size will converge onto the expected probabilistic outcomes. 
So if you had 51% of mining power, there is a chance the other 49% of miners will outcompete you on a short time scale, but over longer time scales we’d expect you to win. 
Really, the threshold for this is just >50%, but a ‘51% attack’ has become the colloquial term for this. 
Note that a 1-2% of an advantage may not sound like much, but that’s the house favour on many casino games, and casinos generally seem to have no problem staying in business (despite sometimes paying out big to lucky players). Over long enough time scales, we reliably see a regression to the mean. 

105482: <p>I thought hashing power is an expression of the probability of being the one to solve the cryptographic puzzle and placing a new block on the network.</p>

<p>If that's true, owning 51% of the hashing power should just give a slightly higher chance of mining a new block than 49% hashing power. However when I read articles explaining Bitcoin and mining, owning 51% of the hashing power in the network is presented as a guarantee to be able to outpace the rest of the network, to the point where an alternate chain can be created and funds double spent.</p>

<p>Given what I assume (could be wrong) is a marginally higher probability of solving the puzzle first at 51%, how does that make sense?
Couldn't someone who got a little lucky do that with 40% of the hashing power? How does 51% of the hashing power guarantee that one will be able to carry out an attack on the network?</p>
 208: <p>If you do not bound the time the attacker has, if he has 51% of the hash power (and is able to hold that majority during the entire attack time), he will eventually always overtake the rest of the network.</p>

<p>More formally: the probability of the attacker overtaking the honest network will converge to 100% as the attack time goes to infinity. There is no guarantee that this happens quickly at all, and that probability will never reach 100%, but it will come arbitrarily close.</p>

<p>Someone with 51% of the hash rate can simply produce blocks faster than the rest of the network combined, so this should not be a surprise. Sure, he may be unlucky for the first block, or the second, or even the first 100 blocks, but in the long term, the rest of the network simply cannot keep up with the attacker.</p>
 63872: <p>With any probability distribution, a larger sample size will converge onto the expected probabilistic outcomes. </p>

<p>So if you had 51% of mining power, there is a chance the other 49% of miners will outcompete you on a short time scale, but over longer time scales we’d expect you to win. </p>

<p>Really, the threshold for this is just >50%, but a ‘51% attack’ has become the colloquial term for this. </p>

<p>Note that a 1-2% of an advantage may not sound like much, but that’s the house favour on many casino games, and casinos generally seem to have no problem staying in business (despite sometimes paying out big to lucky players). Over long enough time scales, we reliably see a regression to the mean. </p>
