The sum of the numbers appearing on two fair dice can range from 2 (when both dice show a 1) to 12 (when both dice show a 6). To determine the probability of each possible outcome, we can use the concept of equally likely outcomes.

Equally Likely Outcomes:
When two dice are rolled, the total number of outcomes is given by the product of the number of outcomes on each die. Since each die has 6 sides, there are a total of 6 * 6 = 36 possible outcomes when two dice are rolled.

Possible Sum Outcomes:
To determine the probability of each possible sum outcome, we can list all the combinations and count how many times each sum appears.

Sum 2: (1, 1)
Sum 3: (1, 2), (2, 1)
Sum 4: (1, 3), (2, 2), (3, 1)
Sum 5: (1, 4), (2, 3), (3, 2), (4, 1)
Sum 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1)
Sum 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)
Sum 8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2)
Sum 9: (3, 6), (4, 5), (5, 4), (6, 3)
Sum 10: (4, 6), (5, 5), (6, 4)
Sum 11: (5, 6), (6, 5)
Sum 12: (6, 6)

Probability Calculation:
To calculate the probability of each sum outcome, we divide the number of favorable outcomes (combinations) by the total number of outcomes.

Probability of Sum 2: 1/36
Probability of Sum 3: 2/36
Probability of Sum 4: 3/36
Probability of Sum 5: 4/36
Probability of Sum 6: 5/36
Probability of Sum 7: 6/36
Probability of Sum 8: 5/36
Probability of Sum 9: 4/36
Probability of Sum 10: 3/36
Probability of Sum 11: 2/36
Probability of Sum 12: 1/36

Highest Probability Outcome:
From the probabilities calculated above, we can observe that the probability is highest for the outcome of 7. The probability of rolling a 7 is 6/36, which simplifies to 1/6 or approximately 0.1667. This is the highest probability among all the possible sum outcomes.

Conclusion:
The highest probability in rolling two fair dice is for the outcome of 7. This is because there are more ways to obtain a sum of 7 (6 combinations) compared to any other sum.