A window contains a ring of six congruent circles, each tangent to its two neighbors and internally tangent to a larger circle, as shown. What is the ratio of the area inside the larger circle that is not covered by the six smaller circles to the area of the larger circle?

1. Let each small circle have radius $r$ and the large circle have radius $R$. The centers of the six small circles are at distance $R-r$ from the center $O$ (each is internally tangent to the large circle).
2. Neighboring small circles are tangent, so adjacent centers are $2r$ apart. The six centers are the vertices of a regular hexagon with center $O$, whose side equals its circumradius: $2r=R-r$, so $R=3r$.
3. The six small circles cover area $6\pi r^2$, while the large circle has area $\pi(3r)^2=9\pi r^2$, so they cover $\frac69=\frac23$ of it.
4. The uncovered fraction is $1-\frac23=\boxed{\dfrac13}$.