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?
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.
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.
There is a guy who has been testing this. He has ran a computer which generates addresses and checks the balances. Out of several years, processing millions of addresses per day, to my knowledge, he has found 3 with small balances. So, you can talk theoretical, where the chances are practically zero, but when there is someone who has attained at least 1, then perhaps it's not as unlikely as the math suggests.
I have been thinking this same question myself. If someone is checking against 100s of public addresses wouldn't that increase the odds of finding a correct private key?
At the same time it checks against 100s of addresses it can effectively dismiss those same addresses. In addition by using what some have noted as not so 100% randomness of private keys made with poor RND generators. If there is even a very slight normal distribution then a person could save more time by concentrating on a certain range of private keys (would still be in the billions).
In fact, what if the person cross-checked with multiple public addresses from different cryptocurrencies? One random seed number producing multiple private keys to check against 10,000s of public address for many cryptocurrencies.
Random Seed 1 used to produce bitcoin, ethereum, monero, ltc, bch, ethereum classic, etc private keys to check against 1000s of public addresses from each cryptocurrency.
Then going to Random Seed 2. And Random Seed 3. Etc.
I am sure that a very resourceful person would find address with non-zero balances much faster than the maths tell us since the math is only looking at 1 private key brute force hacking.
Unfortunatly the chance of finding randomlly using private key is greate , becuase wallets uses almost the same algoritm to generate private keys