Game Theory: Winning with Relatively Prime Numbers

We proceed by finding the probability of winning, and then multiplying it by 1 to get the expected value of the game. There are 9*9=81 possible pairs of numbers that can be chosen. We can count the number of pairs that are relatively prime using complementary counting. There are 4 even numbers between 2 and 10, so there are 5 odd numbers that can be chosen. Out of these, 2, 4, 6, and 8 are not relatively prime with 4, while 4 can be paired with 3, 5, 7, and 9. Therefore, there are 4 successful pairings for every 5 pairings, so the probability of winning is 4/5.

Thus, the value of the game is 4/5.