I'm a little bit confused here, and quite upset about bugging on such simple probabilistic problem !!

Since we are supposed to pick the 2nd ball in the same bag we picked the first one, which was black, there are 2 possible outcomes:

Either I picked a black ball from the bag with 2 black balls, and in that case the 2nd will be black, or I picked the black ball from the bag which contained a black and a white ball, and in that case I'll pick a white ball.

So the probability of picking a black ball, given that I have to pick

__from the same bag__, should be 1/2.

It would be 2/3 if I would have the choice of picking the 2nd ball from any of the 2 bags.

What did I miss here ?