The first 1 comes from the address version and is always there for a normal bitcoin address. Every other 1 requires a byte equal to 0 is the pub key hash.
- 11: 1/2^8 = 1/256
- 111: 1/2^16 = 1/65536
- 1111: 1/2^24
etc
are you sure?
It seems base58 would have a 1/58th chance for a '1'