Game Theory: Probability of Winning with Relatively Prime Numbers

Let's first count the total number of possible outcomes. Each player can choose from 9 possible numbers (2 through 10, excluding the number chosen by the other player). Therefore, there are a total of 9 x 9 = 81 possible outcomes.

Now, we need to count the number of outcomes where the two numbers are relatively prime. We can do this by first counting the number of outcomes where the two numbers are not relatively prime, and then subtracting that from the total number of outcomes.

If the two numbers have a common factor of 2, then the only possible choices are (2, 3), (2, 5), (2, 7), and (2, 9), for a total of 4 outcomes. If the two numbers have a common factor of 3, then the only possible choices are (3, 4), (3, 5), (3, 7), and (3, 8), for a total of 4 outcomes. If the two numbers have a common factor of 5, then the only possible choices are (5, 6) and (5, 8), for a total of 2 outcomes. If the two numbers have a common factor of 7, then the only possible choice is (7, 8), for a total of 1 outcome. If the two numbers have a common factor of 9, then the only possible choice is (9, 4), for a total of 1 outcome.

Therefore, there are a total of 4 + 4 + 2 + 1 + 1 = 12 outcomes where the two numbers are not relatively prime. So, the number of outcomes where the two numbers are relatively prime is 81 - 12 = 69.

Since each outcome is equally likely, the probability of the two numbers being relatively prime is 69/81 = 2/3. Therefore, the value of the game is 2/3.