Step 1: Organise Outcomes by the First Die Value:
Instead of grouping by the sum first, go through each possible value of the first roll from 1 to 6 in turn, and for each fixed first roll value, count how many of the six possible second roll values produce a total that is a prime number.
Step 2: Count Favourable Second-Roll Values for Each First-Roll Value:
If the first roll is 1, the possible sums with second roll 1 through 6 are 2, 3, 4, 5, 6, 7; the prime sums among these are 2, 3, 5 and 7, coming from second roll values 1, 2, 4 and 6, giving 4 favourable outcomes.
If the first roll is 2, the possible sums are 3, 4, 5, 6, 7, 8; the prime sums are 3, 5 and 7, coming from second roll values 1, 3 and 5, giving 3 favourable outcomes.
If the first roll is 3, the possible sums are 4, 5, 6, 7, 8, 9; the prime sums are 5 and 7, coming from second roll values 2 and 4, giving 2 favourable outcomes.
If the first roll is 4, the possible sums are 5, 6, 7, 8, 9, 10; the prime sums are 5 and 7, coming from second roll values 1 and 3, giving 2 favourable outcomes.
If the first roll is 5, the possible sums are 6, 7, 8, 9, 10, 11; the prime sums are 7 and 11, coming from second roll values 2 and 6, giving 2 favourable outcomes.
If the first roll is 6, the possible sums are 7, 8, 9, 10, 11, 12; the prime sums are 7 and 11, coming from second roll values 1 and 5, giving 2 favourable outcomes.
Step 3: Add Up the Favourable Outcomes:
Adding the favourable counts across all six first roll values gives 4 + 3 + 2 + 2 + 2 + 2 = 15 favourable outcomes out of the 36 equally likely ordered pairs.
Step 4: Final Answer:
Dividing the favourable outcomes by the total outcomes gives the same probability as before.
\[ oxed{\dfrac{15}{36}} \]