This is just a matter of curiosity. I know a woman who is Jewish and has an adopted daughter. The daughter's biological mother was not Jewish. The daughter was raised Jewish so my question would be, from a strict interpretation, is she considered Jewish? I suspect she would be equivalent to someone who has gone through conversion to Judaism but was just curious to find out for sure.
Part two of my question is when she gets older and has a child of her own, if that child decides to not be a practicing Jew, would he or she be considered Jewish?
Was she converted as a child when the mother adopted her? If so, when she becomes bat mitzvah, she can decide to accept or reject the conversion, since as a child she didn't have a choice. Once she reaches that age and still keeps doing mitzvot and Jewish custom, then it is considered her acceptance of the conversion, and she would be a Jew who is transgressing if she decides not to keep halacha or violates certain precepts at a later time. But if she does not keep mitzvot from when she becomes a bat mitzvah (age 12), then she is considered still a non-Jew, considered that she rejected the conversion and never became a Jew in the first place, and would not be transgressing when in the rest of her life she does not keep Jewish law.
However, with regards to a plain adoption where there was not a conversion process (dunking in mikve), I'm not sure, I'll have to ask my rabbi. I imagine that if she is not sure, she can do a conversion le'chumrah to remove any doubt.