When I was arrested for possession of drugs at the age of 19 can I get my record expunged when I am 21 now.

Depdnds on the State but there is often a 5 to 10 year waiting time. The record, by the way, would still be visibile to Police Agencies.

What State?

Ohio

A lot will depend on where you live, in some states ( in the US) you can't explunge records, in others if you were sentenced under first offender rules, you can do it early. But on average it is about 8 years after you finish your sentence for the crime.

You will have to wait for at least 10 years, be arrest free and have this presented to the court by an attorney. It is not immediately granted or denied either and can take up to 9-10 months as well.