I understand the probability, but say you have all your savings on an address, and, god forbid, someone else "mints" an address that has the same public key, isn't that really HORRIBLE?!
You could also die tomorrow getting eaten by a shark while struck by lightning in a desert. The odds of that are probably higher than a 160 bit collision. You could also buy one ticket, win the powerball lottery jackpot, next day buy a ticket win that powerball lottery jackpot, next day buy another ticket win that powerball lottery jackpot, next day buy a ticket, win that jackpot, next day buy a ticket, buy that jackpot, next day buy a ticket, win that jackpot, next day buy a ticket, win that jackpot. The odds of doing that is more probably than generating a 160bit collision.
the public key is [0-9a-zA-Z] so its not really 2^160 right? It's 32 characters with 10+28+28 mutex no?
Well not exactly. The address isn't the public key. The address is a checksummed hash of the public key.
So in Bitcoin you have:
* private key - 256 bit number
* public key - 512 bit ECDSA (x,y pair) -essentially a number
* public address - 200 bits (8 bit version identifier) + (160 bit hash of public key) + (32 bit checksum - to prevent sending funds to invalid addresses)
Public addresses are simply 200 bit numbers. We put them into Base58 format to make them human readable but they are just numbers to the network.
All three of the following numbers are the same and represent a bitcoin address, they are just in different formats.
00010966776006953D5567439E5E39F86A0D273BEED61967F6 (in hexadecimal)
0110011101110110000000000110100101010100000000000000000000000000
0110101000001101001001110000000000000000000000000000000000000000 (in binary)
16UwLL9Risc3QfPqBUvKofHmBQ7wMtjvM (in Base58 format)
The Base58 is the most human readable (and compact) but all three formats have the same value.
Someone made this awesome graphic to show the details
More details:
https://en.bitcoin.it/wiki/Technical_background_of_Bitcoin_addresses