A mathematical proof method with details for "near to zero chance of generating the same pair key wallet" I can not find a clear mathematical proof for the top sentence with example
According to https://en.bitcoin.it/wiki/Private_key
any 256-bit number from 0x1 to 0xFFFF FFFF FFFF FFFF FFFF FFFF FFFF FFFE BAAE DCE6 AF48 A03B BFD2 5E8C D036 4140 is a valid private key.
So if someone generates a private key using the usual random process, and then you generate a private key using a random process there is one chance in 0xFFFF FFFF FFFF FFFF FFFF FFFF FFFF FFFE BAAE DCE6 AF48 A03B BFD2 5E8C D036 4140 that you will generate the same key.
If my workings are correct, that is one chance in 115792089237316195423570985008687907852837564279074904382605163141518161494336 or one chance in around 1.158 x 1077
In terms of a probability, this is fairly close to zero. It is around 0.00000000000000000000000000000000000000000000000000000000000000000000000000000864
It is hard to grasp the scale of these numbers. According to https://www.universetoday.com/36302/atoms-in-the-universe/
it is estimated that the there are between 1078 to 1082 atoms in the known, observable universe
Related questions that might be informative
- Is each Bitcoin address unique?
- How long would it take to brute force an AES-128 key? (Bitcoin doesn't use AES but the answers give some idea of the scales involved)