I guess you didn't watch the first couple of episodes. In those they introduce Detective Reese as someone who's had problems - I think it was booze, drinking on the job or something (it was vague). She had to go to AA or else lose her job and career. The series starts with her back on the job after her mandatory AA sessions. It was because of her problem that she had poor stature in the department and got the 'crappy' assignment of being partnered with Detective Crews, who no one wanted to be partnered with (including her for most of Season 1; doing a fantastic job of playing the frustrated partner, always aggravated at Crews but slowly learning to respect him). Anyway, it's because Reese is familiar with xxx-anonymous - their meeting weekly (seen on the calendar), etc. - that she recognized that LWA could be one of those types of groups. So you were right on to recognize that only a recovering addict would spot that. But nothing to lol about, within the context of the characters.

Lottery Winners Anonymous rofl Where do they pick up these writers...