Two fair six-sided dice are rolled. What is the probability that the total score is either 5 or 6?

Introduction / Context:
In this problem two fair dice are rolled and we are interested in the event that the total score is either 5 or 6. This combines the probabilities of two different sums using the idea of mutually exclusive events. It tests the ability to count combinations for specific sums and to add probabilities correctly.

Given Data / Assumptions:

• Two fair six-sided dice are rolled simultaneously.
• Each die shows an integer from 1 to 6.
• There are 36 equally likely ordered outcomes.
• We require the probability that the sum is 5 or 6.

Concept / Approach:
We first count the number of ordered pairs that sum to 5 and then the number that sum to 6. Since the events "sum is 5" and "sum is 6" are mutually exclusive (they cannot happen together on the same trial), the total favourable outcomes are the sum of these counts. Finally, we divide by 36 to get the probability.

Step-by-Step Solution:
Number of ordered pairs where the sum is 5: Pairs: (1,4), (2,3), (3,2), (4,1). So there are 4 outcomes with sum 5. Number of ordered pairs where the sum is 6: Pairs: (1,5), (2,4), (3,3), (4,2), (5,1). So there are 5 outcomes with sum 6. Total favourable outcomes (sum 5 or 6) = 4 + 5 = 9. Total possible outcomes from two dice = 6 * 6 = 36. Required probability = 9 / 36. Simplify 9 / 36 by dividing numerator and denominator by 9 to get 1 / 4.

Verification / Alternative check:
Using the known distribution of sums for two dice, the counts for sums 5 and 6 are 4 and 5 respectively. Adding them gives 9. Dividing 9 by 36 leads again to 1/4. This matches the explicit enumeration and confirms that the probability is 0.25.

Why Other Options Are Wrong:
2/14: This fraction simplifies to 1/7, which is smaller than the correct value and does not align with the count of favourable outcomes.
5/18: This would correspond to 10 favourable outcomes (because 5/18 of 36 equals 10), which is incorrect.
3/4: This is much too large and would imply that most outcomes produce a sum of 5 or 6.
None of these: This is incorrect because 1/4 is a listed option and is correct.

Common Pitfalls:
Some learners miscount the number of ways to get a particular sum, especially by forgetting that (2,3) and (3,2) are distinct outcomes. Another common error is to average the probabilities of sum 5 and sum 6 rather than adding them, which gives a completely wrong result. Always count ordered pairs carefully and remember that mutually exclusive events have their probabilities added, not averaged.

Final Answer:
Therefore, the probability that the total score is 5 or 6 is 1/4.