Conditioned Dice Sum Expectation and Probability

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Conditional Expectation.

Your Task
Two fair 6-sided dice are rolled independently. Let $S = X_1 + X_2$ be the sum of their values. Given that the sum is strictly greater than 7, what is the conditional expectation $\mathbb{E}[S \mid S > 7]$ and the conditional probability that at least one die shows a 6?

Constraints

• All trials are mutually independent and identically distributed.
• Probabilities lie strictly within $(0, 1)$ unless degenerate.