Two unbiased dice thrown together: What is the probability that the sum of the two faces is divisible by 3?

Introduction / Context:
We classify die faces by residues modulo 3 and use independence. A 1–6 die has two faces of each residue class 0,1,2 (since 3 and 6 are 0 mod 3; 1,4 are 1 mod 3; 2,5 are 2 mod 3).

Given Data / Assumptions:
Die outcomes are independent and uniform on {1,…,6}.

Concept / Approach:
A sum is divisible by 3 when residues add to 0 mod 3: (0,0), (1,2), or (2,1). Each residue probability per die is 2/6.

Step-by-Step Solution:

Verification / Alternative check:
Explicit counting yields 12 favorable ordered pairs out of 36 total, confirming 1/3.

Why Other Options Are Wrong:
11/36 and 13/36 are off by ±1 pair; 17/36 is too large.

Common Pitfalls:
Forgetting that there are two faces in each residue class; double-counting permutations.

Final Answer:
12/36