Solution:

The probability that a contestant gets all $3$ correct is $$\frac 13^3 = \frac 1{27}.$$ The probability that a contestant gets exactly $2$ is $$\frac 13^2\cdot \frac 23 \cdot \binom {3}{2} = \frac 6{27}.$$ The combined probability is $$\frac{6}{27} + \frac 1{27} = \frac {7}{27}.$$

Thus, the correct answer is D .