If Polya is never to visit the sixth floor after she begins, we know that her first stop is at the seventh floor. Moreover, her second stop must be at the eighth floor. She has three moves left, and the only way she can ever visit the 6th floor from the 8th floor in three remaining moves is to go down in both of the following two steps. The probability of getting to the eighth step in two moves is $ \frac{1}{2^2} = \frac{1}{4} $. And the probability of not going downward in the next two steps is $ 1- \frac{1}{4} = \frac{3}{4} $. So the overall probability of never hitting the sixth floor after the beginning is $ \frac{1}{4} \cdot \frac{3}{4} = \boxed{\frac{3}{16}} $.