Two fair dice are thrown together. What is the probability that the sum of the numbers appearing on the two faces is divisible by 4 or divisible by 6?

Introduction / Context:
This question tests understanding of probability on two dice where the event is expressed in terms of the sum being divisible by given numbers. It requires identifying all possible sums of two dice and determining for which of those sums divisibility by 4 or 6 holds, including handling overlap correctly.

Given Data / Assumptions:

• Two fair six sided dice are thrown.
• Each ordered pair of outcomes (1 to 6 on each die) is equally likely.
• We are interested in sums divisible by 4 or by 6.
• Total number of possible outcomes is 6 * 6 = 36.

Concept / Approach:
First list all possible sums of two dice, which range from 2 to 12. Identify sums divisible by 4 or 6. For each such sum, count the number of ordered pairs giving that sum. Because we work at the level of pairs, some sums have more combinations than others. The desired probability is the number of favourable pairs divided by 36.

Step-by-Step Solution:
Possible sums: 2 to 12. Sums divisible by 4: 4, 8, 12. Sums divisible by 6: 6, 12. List combinations: Sum 4: (1, 3), (2, 2), (3, 1) so 3 ways. Sum 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) so 5 ways. Sum 8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) so 5 ways. Sum 12: (6, 6) so 1 way. Total favourable outcomes = 3 + 5 + 5 + 1 = 14. Probability = 14 / 36 = 7 / 18 after simplification.

Verification / Alternative check:
Instead of listing pairs, we can think in terms of the frequency table of sums. The counts 3, 5, 5, and 1 for sums 4, 6, 8, and 12 are standard values for two dice and sum to 14. Dividing 14 by 36 again gives 7/18, confirming the answer.

Why Other Options Are Wrong:
14/35 reduces to 2/5, which would correspond to 14 or 15 favourable out of 35, not out of 36. 8/18 simplifies to 4/9 and corresponds to 16 favourable outcomes, which is not the case. 7/35 is far too small and would match 7 favourable outcomes, which is incorrect.

Common Pitfalls:
Common mistakes include treating all sums from 2 to 12 as equally likely, ignoring their different frequencies, or double counting sums like 12 which satisfy both divisibility conditions. The safest method is always to count ordered pairs when working with two dice problems.

Final Answer:
Thus, the probability that the sum is divisible by 4 or by 6 is 7/18.