Game Theory: Finding the Expected Value of a Number Game

We can use complementary counting to find the probability that the numbers are not relatively prime. The only way for the numbers to not be relatively prime is if they share a common factor of 2, 3, 5, or 7.

The probability that both numbers are even is $\frac{4}{9} \cdot \frac{4}{9} = \frac{16}{81}$. However, we have overcounted the case where both numbers are multiples of 4. The probability that both numbers are multiples of 4 is $\frac{1}{9} \cdot \frac{1}{9} = \frac{1}{81}$. Similarly, we have overcounted the cases where both numbers are multiples of 6 or 10.

Using the Principle of Inclusion-Exclusion, the probability that the numbers are not relatively prime is:

$$\frac{16}{81} - \frac{4}{81} - \frac{4}{81} - \frac{4}{81} + \frac{1}{81} + \frac{1}{81} + \frac{1}{81} = \frac{11}{27}$$

Therefore, the probability that the numbers are relatively prime is $1 - \frac{11}{27} = \frac{16}{27}$.

Since the game awards 1 score for relatively prime numbers, the expected value of the game is:

$$\frac{16}{27} \cdot 1 + \frac{11}{27} \cdot 0 = \boxed{\frac{16}{27}}$$