Two fair six-sided dice are tossed together. What is the probability that the total score shown is a prime number?

Introduction / Context:
This question examines the probability that the sum of two fair dice is a prime number. It is a standard problem in aptitude tests and helps reinforce knowledge of prime numbers, enumeration of outcomes, and calculation of probability from a finite sample space.

Given Data / Assumptions:

• Two fair six-sided dice are rolled simultaneously.
• Each die has faces numbered 1 to 6.
• There are 36 equally likely ordered outcomes (6 * 6).
• We want the probability that the total score (sum of the two faces) is a prime number.

Concept / Approach:
Prime numbers possible as sums of two dice range from 2 to 12. We first list the prime numbers in this range and then count the number of ordered pairs (i, j) that add up to each prime. The probability is the ratio of the total number of favourable ordered pairs to the total of 36 outcomes.

Step-by-Step Solution:
Possible sums from two dice range from 2 to 12. Prime numbers in this range are: 2, 3, 5, 7, 11. Sum 2: (1,1) gives 1 outcome. Sum 3: (1,2), (2,1) gives 2 outcomes. Sum 5: (1,4), (2,3), (3,2), (4,1) gives 4 outcomes. Sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) gives 6 outcomes. Sum 11: (5,6), (6,5) gives 2 outcomes. Total favourable outcomes = 1 + 2 + 4 + 6 + 2 = 15. Total possible outcomes = 6 * 6 = 36. Required probability = 15 / 36 = 5 / 12 after dividing numerator and denominator by 3.

Verification / Alternative check:
We can verify by observing that the sum distribution of two dice is symmetric and well known. The counts for sums 2 to 12 are 1,2,3,4,5,6,5,4,3,2,1. Selecting only the prime sums 2,3,5,7,11 and adding their counts gives 1 + 2 + 4 + 6 + 2 = 15, which is consistent with the previous enumeration. Dividing by 36 again yields 5/12, confirming the result.

Why Other Options Are Wrong:
1/6: This would correspond to only 6 favourable outcomes, which does not match the actual count of 15.
1/2: This implies 18 favourable outcomes, which is larger than the actual value.
3/4: This is much too large and would mean that three quarters of all outcomes give a prime sum, which is not true.
None of these: This is incorrect because 5/12 is present and correct.

Common Pitfalls:
Many learners forget that the dice are ordered, so (1,2) and (2,1) are two different outcomes. Another mistake is misidentifying prime numbers or including 1 as a prime. It is also easy to miscount the number of combinations that give each sum if one does not list them systematically. Careful enumeration avoids these errors.

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