## Coin flipping game

Your friend challenges you to a game in which you flip a fair coin until you get heads. If you flip an even number of times, you win. Let $$A$$ be the event that you win. Let $$B$$ be the event that you flip the coin 3 or more times. Let $$C$$ be the event that you flip the coin 4 or more times.

1. Compute $$\mathbb{P}(A)$$.
2. Are $$A$$ and $$B$$ independent?
3. Are $$A$$ and $$C$$ independent?

## Repeated Quiz Questions

Each week you get multiple attempts to take a two-question quiz. For each attempt, two questions are pulled at random from a bank of 100 questions. For a single attempt, the two questions are distinct.

1. If you attempt the quiz 5 times, what is the probability that within those 5 attempts, you’ve seen at least one question two or more times?
2. How many times do you need to attempt the quiz to have a greater than 50% chance of seeing at least one question two or more times?

## Dice Rolling Events

Consider rolling a fair 6-sided die twice. Let $$A$$ be the event that the first roll is less than or equal to 3. Let $$B$$ be the event that the second roll is less than or equal to 3. Find an event $$C$$ in the same outcome space as $$A$$  and $$B$$ with $$0<\mathbb{P}(C)<1$$ and  such that $$A$$, $$B$$ and $$C$$ are mutually independent, or show that no such event exists.