This problem is cited by M. Gardner in his Mathematical Circus and also Gardner's Workout.
As the state of the bag, after the operation, is necessarily identical with its state before it, the chance is just what it was, viz. 1/2.Solution #2
Let B and W1 stand for the black or white counter that may be in the bag at the start and W2 for the added white counter. After removing white counter there are three equally likely states:
|Inside bag||Outside bag|
In two of these states a white counter remains in the bag, and so the chance of drawing a white counter the second time is 2/3.