7. In a bag, there are two black and two white Go stones. Now, stones are taken from the bag, one at a time. If a white stone is drawn, it is kept. If a black stone is drawn, a white stone must be put back. If there are no white stones left to put back, the game ends. What is the probability of successfully taking all the stones? $\qquad$
7. $\frac{7}{18}$.

The successful scenarios for removing all the pieces can be divided into two categories.
(1) Drawing two white pieces consecutively, with a probability of $P_{1}=\frac{1}{2} \times \frac{1}{3}=\frac{1}{6}$.
(2) Drawing a white piece first, then a black piece. In this case, the third draw must be a white piece (note that there are three pieces left when drawing the third piece). The probability of this occurring is $P_{2}=\frac{1}{2} \times \frac{2}{3} \times \frac{2}{3}=\frac{2}{9}$. Therefore, the required probability is $P_{1}+P_{2}=\frac{1}{6}+\frac{2}{9}=\frac{7}{18}$.