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.
- Compute \(\mathbb{P}(A)\).
- Are \(A\) and \(B\) independent?
- 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.
- 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?
- 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.
