Two fair dice are tossed simultaneously. What is the probability that the total score shown on the two upper faces is a prime number?

Introduction / Context:
This question checks familiarity with basic probability for two dice and understanding of prime numbers. The key idea is to list all possible sums when two dice are thrown and then count how many of those sums are prime numbers, while remembering that some sums occur more frequently than others.

Given Data / Assumptions:

• Two standard fair six sided dice are thrown at the same time.
• Each die shows an integer from 1 to 6.
• All 36 ordered pairs of outcomes are equally likely.
• Prime numbers to consider among possible sums are 2, 3, 5, 7 and 11.

Concept / Approach:
For two dice, there are 6 * 6 = 36 total possible ordered outcomes. For each possible sum, we count how many pairs (first die, second die) produce that sum. The probability that the sum is prime is the number of favourable outcome pairs divided by 36. We do not treat all sums as equally likely; we treat all ordered pairs as equally likely.

Step-by-Step Solution:
Total number of outcomes = 6 * 6 = 36. Sum = 2: only (1, 1) so 1 way. Sum = 3: (1, 2), (2, 1) so 2 ways. Sum = 5: (1, 4), (2, 3), (3, 2), (4, 1) so 4 ways. Sum = 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) so 6 ways. Sum = 11: (5, 6), (6, 5) so 2 ways. Total favourable outcomes = 1 + 2 + 4 + 6 + 2 = 15. Probability = 15 / 36 = 5 / 12 after simplification.

Verification / Alternative check:
We can verify by writing the full 6 by 6 table of sums, circling those that are 2, 3, 5, 7 or 11 and counting circles. This again produces 15 favourable pairs out of 36. Dividing numerator and denominator of 15/36 by 3 gives 5/12, confirming the final fraction is simplified.

Why Other Options Are Wrong:
1/6: This is too small and would correspond to only 6 favourable cases. 1/2: This would require 18 favourable outcomes, which is more than the actual 15. 7/9: This is greater than 3/4 and suggests 28 favourable outcomes, which is impossible here.

Common Pitfalls:
Learners sometimes count prime sums but then mistakenly divide by 11 possible sums instead of 36 outcome pairs. The sample space in dice problems is the set of ordered pairs, not merely the set of distinct sums, which have different frequencies. Forgetting that, leads to incorrect probabilities.

Final Answer:
Hence, the probability that the total score is a prime number is 5/12.