Ask yourself why you feel any need to tell your roommate. Unless you are expecting them to "support you" in some way, this information is completely irrelevant to your relationship.
Typically, roommates (I've lived with several and live with one now) aren't going to query you about your whereabouts except in a very casual, conversational way. "You heading out?" Just answer, "Gotta run a few errands. See you later."
As long as you're working your program and handling your business, I don't think it's even appropriate to "dump" this info on your roomie. If they're a "normie," they don't know much about alcoholism and should have the right to not be worried about it. To be comfortable living their normal lifestyle under their own roof without having awkward knowledge of a roommate's prior alcohol abuse.
Bottom line: If you're stable, no need to tell. If you're not, you probably shouldn't be rooming with a normie.