Rapid-Fire Probability Questions

Context

Answer the following independent probability questions. Assume fair dice/coin and a standard 52-card deck. Show formulas or brief reasoning.

Questions

1. Two fair six-sided dice are rolled. What is the probability distribution of the sum?
2. From a well-shuffled standard 52-card deck, after discarding the top 10 cards unseen, what is the probability that the next card is red?
3. When rolling two fair dice, what is the probability that the sum is 11 or 12?
4. In a fair-coin gambler’s-ruin process, starting with bankroll 10 and goal 20 (absorbing barriers at 0 and 20, betting 1 each round), what is the probability of reaching 20 before 0?
5. If a fair six-sided die is rolled N times, what is the expected number of occurrences of each face and the variance of that count?
6. If four fair six-sided dice are rolled, what is the probability that the product of the outcomes is odd?