Game Theory: Expected Value of Relatively Prime Numbers

We can solve this problem by finding the probability that the two numbers are relatively prime. Let's consider all the possible pairs of integers that can be written between 2 and 10 (inclusive). There are 9 choices for the first number and 9 choices for the second number, for a total of 81 possible pairs.

To count the number of pairs that are relatively prime, we can use the principle of inclusion-exclusion. Let A be the set of pairs with a common factor of 2, B be the set of pairs with a common factor of 3, and C be the set of pairs with a common factor of 5 or 7. We want to count the number of pairs that are not in any of these sets.

The number of pairs in set A is 44=16, since there are 4 even numbers between 2 and 10 and any two even numbers have a common factor of 2. Similarly, the number of pairs in set B is 22=4, since there are 2 multiples of 3 and any two multiples of 3 have a common factor of 3. Finally, the number of pairs in set C is 2*2=4, since there are 2 multiples of 5 and 2 multiples of 7, and any pair of numbers with a common factor of 5 or 7 must include one of these multiples.

To count the number of pairs that are in sets A and B, we can choose one of the 2 multiples of 3 and then choose any of the 2 even numbers that are not multiples of 3. This gives us 2*2=4 pairs. Similarly, the number of pairs in sets A and C is 4, and the number of pairs in sets B and C is also 4.

To count the number of pairs that are in all three sets, we can choose one of the 2 multiples of 3, one of the 2 multiples of 5, and one of the 2 multiples of 7. This gives us 222=8 pairs.

Using the principle of inclusion-exclusion, we can count the number of pairs that are not in any of the sets as follows:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C| |A ∪ B ∪ C| = 16 + 4 + 4 - 4 - 4 - 4 + 8 |A ∪ B ∪ C| = 20

Therefore, there are 20 pairs of integers between 2 and 10 that are not relatively prime. The probability that a randomly chosen pair is not relatively prime is therefore 20/81. The probability that a randomly chosen pair is relatively prime is 1 - 20/81 = 61/81.

Since you win 1 score if the numbers are relatively prime, the expected value of your score is:

E(score) = (1)(61/81) + (0)(20/81) E(score) = 61/81

Therefore, the value of the game is 61/81.