Solution:

Answer: $\frac{11}{16}$

Let the starting vertex be the 'bottom' one. Then there is a 'top' vertex, and 4 'middle' ones. If $p(n)$ is the probability that Sherry is on a middle vertex after $n$ minutes, $p(0)=0$,
$p(n+1) = (1 - p(n)) + p(n) \cdot \frac{1}{2}$.

This recurrence gives us the following equations.
$$\begin{aligned} p(n+1) & = 1 - \frac{p(n)}{2} \\ p(0) & = 0 \\ p(1) & = 1 \\ p(2) & = \frac{1}{2} \\ p(3) & = \frac{3}{4} \\ p(4) & = \frac{5}{8} \\ p(5) & = \frac{11}{16} \end{aligned}$$