Fair die rolled twice: What is the probability the die shows the same number on both rolls (i.e., a double)?

Introduction / Context:On two independent fair die rolls, there are 36 ordered outcomes. We want those where both faces match (1–1, 2–2, …, 6–6).

Given Data / Assumptions:Uniform outcomes on pairs (1,…,6) × (1,…,6).

Concept / Approach:Count matching pairs and divide by total pairs.

Step-by-Step Solution:

Verification / Alternative check:Alternatively, P(match) = sum over i of P(first=i)*P(second=i) = 6*(1/6)*(1/6) = 1/6.

Why Other Options Are Wrong:1/12 or 1/24 undercount; 1/8 corresponds to 4.5 favorable outcomes (nonsense).

Common Pitfalls:Confusing ordered vs unordered outcomes; both methods agree on 1/6.

Final Answer:1/6