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Say I have created a paper wallet with one bitcoin address in a vault. What are chances that someone else will find the private key for this wallet?

Related: what are the chances of vanitygen address - private key pair collision?

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Essentially zero. They would have to find a private key whose public key hash matched yours. The public key hash is 160 bits long. If they had a billion computers, each of which could try a billion keys a second, and they tried for a billion years, they'd have much, much less than a one in a billion chance of getting it.

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I understand this if you're trying to match a single public key. But with the entire blockchain freely available and the entire world possibly working on checking each private key, constantly on thousands of different computers, the chances should be slightly different than the model you described. –  Luca Matteis Feb 4 at 13:22
    
I assumed a billion computers, and I gave a billion years. There are only 21 million Bitcoins. So if you figure it's not worth much trouble to steal less than 0.01 Bitcoin, that means at most there are 2 billion keys worth stealing. That doesn't change the numbers much (when you make the number of computers and number of years reasonable). –  David Schwartz Feb 4 at 13:24
    
There are 21 million bitcoins yes, but many more addresses. –  Luca Matteis Feb 4 at 13:25
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Since there are 21 million Bitcoins, that means at most 2.1 billion keys could hold at least 0.01 Bitcoins. –  David Schwartz Feb 4 at 13:27
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There are around 1,461,501,637,330,902,918,203,684,832,716,283,019,655,932,542,976 posible bitcoin addresses according to BitcoinTalk.org. This makes it a very small posibility of finding another address that is being used. So the chance is around %0.

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