Probes
In 1998, the Norwegian Supreme Court ruled that scanning/probing
a system is not by itself illegal. The ruling is based upon much
the same legal and technical basis that you might find in other
countries.
Right now in the United States, there are few laws on the books
that regulate such things. Not only are the laws deficient, there
are some technology problems as well. The biggest hurdle is
IP spoofing,
which would allow a hacker to send scans at your system while
pretending to be somebody else (it isn't useful to the hacker
other than getting somebody else in trouble).
In summary, chances are small that you could prosecute a hacker
for scanning your system.
Break-in
If a hacker successfully breaks into your computer, you could
go after the hacker. However, the hacker can come after your
machine from anywhere in the world; most countries have laws
against hacking, but trying to prosecute them from far away
is very difficult.
The police will not want to get involved unless there is
a major financial loss involved. In order to get them involved,
it helps to have air-tight evidence. Getting such evidence
can be very difficult. One way to start the process is to have
a complete packet log of everything the hacker did.
In addition, you will need to personally harass the ISP
where the hacker came from in order to find out who had that
IP address during that time.
