1. **Problem Statement:** A student rolls a die repeatedly until the first "4" appears. Define $X$ as the number of rolls needed to get this first "4" (including the roll that shows "4"). After this, the student continues rolling until the first "3" appears. Define $Y$ as the number of rolls after the first "4" up to and including the roll that shows "3". We want to determine if $X$ and $Y$ are independent random variables. 2. **Understanding the problem:** - $X$ counts rolls until the first "4". - $Y$ counts rolls after the first "4" until the first "3". 3. **Key properties and formulas:** - Each roll of a fair die is independent. - The probability of rolling any specific number (like "4" or "3") is $\frac{1}{6}$. - The number of rolls until a specific outcome in independent trials follows a geometric distribution. 4. **Distribution of $X$ and $Y$:** - $X \sim \text{Geometric}(p=\frac{1}{6})$ with $P(X=k) = \left(\frac{5}{6}\right)^{k-1} \cdot \frac{1}{6}$ for $k=1,2,3,...$ - Similarly, $Y \sim \text{Geometric}(p=\frac{1}{6})$ with the same probability mass function. 5. **Are $X$ and $Y$ independent?** - The rolls before the first "4" do not affect the rolls after the first "4" because the die rolls are independent. - The process "resets" after the first "4"; the count for $Y$ depends only on rolls after that point. - Therefore, $X$ and $Y$ are independent random variables. 6. **Justification:** - Independence means $P(X=x, Y=y) = P(X=x) \cdot P(Y=y)$ for all $x,y$. - Since the die rolls are independent and memoryless, the waiting times for "4" and then for "3" are independent geometric random variables. **Final answer:** $X$ and $Y$ are independent random variables because the number of rolls until the first "4" does not influence the number of rolls after the first "4" until the first "3" due to the independence and memoryless property of the die rolls.