Let each smallest circle have radius $1.$

This means that there are $3$ shaded unit circles, which total to $3 \cdot 1^2 \pi = 3 \pi$ area.

There is also a shaded circle with radius $4$ with two unshaded circles of radius $2$ inside.

This gives us an extra shaded area of $$\begin{align*} 4^2 \pi - 2 \cdot 2^2 \pi &= 16 \pi - 8 \pi \\ &= 8 \pi. \end{align*}$$

Adding these values together yields $$8\pi + 3\pi = 11\pi.$$ The area of the large unshaded circle is $$6^2\pi = 36\pi.$$ Therefore, the desired fraction is $$\dfrac{11\pi}{36\pi} = \dfrac{11}{36}.$$

Thus, B is the correct answer.