jmc

The probability for a single dart to land in the non-shaded region is the ratio of the area of the non-shaded region to the area of the entire dartboard. The area of the entire dartboard is $\pi \cdot 6^2=36\pi$. The area of the shaded region is the area of the second largest circle minus the area of the smallest circle, or $\pi \cdot 4^2-\pi \cdot 2^2=12\pi$, so the area of the non-shaded region is $36\pi -12\pi =24\pi$. Thus, our ratio is $\frac{24\pi }{36\pi }=\frac{2}{3}$. If each dart has a $\frac{2}{3}$ chance of landing in a non-shaded region and there are 9 darts, then the expected number of darts that land in a non-shaded region is $9\cdot \frac{2}{3}=\boxed{6}$.