12. A bag contains 8 white balls and 2 red balls. Each time a ball is randomly drawn, and then a white ball is put back. The probability that all red balls are exactly drawn by the 4th draw is $\qquad$
Let $A$, $B$, and $C$ represent the events of drawing a red ball on the first, second, and third draws, respectively, and the event that the fourth draw exactly completes all the red balls. Then, $P(A)=\frac{1}{5} \times \frac{9}{10} \times \frac{9}{10} \times \frac{1}{10} ; P(B)=\frac{4}{5} \times \frac{1}{5} \times \frac{9}{10} \times \frac{1}{10}$; $P(C)=\frac{4}{5} \times \frac{4}{5} \times \frac{1}{5} \times \frac{1}{10}$. Thus, $D=$ “the fourth draw exactly completes all the red balls”, $P(D)=P(A)+P(B)+P(C)=\frac{162+144+128}{10^{4}}=\frac{217}{5000}$