\Question
\textbf{Strange Dilution}
You have a jar of red marbles and blue marbles. At each time step, you draw a marble, and you note the color of the marble. Then, you dilute the proportion of the opposite-colored marbles by a factor of $\gamma$, where $0 < \gamma < 1$. (For example: if you pick a red marble, then the proportion of blue marbles is reduced by a factor of $\gamma$.) If $p$ is the fraction of marbles that started off as red, what is the expected proportion of red marbles at time $n$?
\Question \textbf{A meeting of three}
James, Jon, and Mike each arrive at the movie theater at times uniformly distributed in the interval $(3:00, 3:20)$. Their arrival times are independent. Each person waits 5 minutes after their arrival before heading into the theater. What is the probability they all see each other before going into the theater?
\Question \textbf{Exponential Median}
What is the expected value of the median of three i.i.d exponential variables with parameter $\lambda$?
\Question \textbf{Random walk}
Alice starts at vertex 0 and wishes to get to vertex $n$. When she is at vertex 0 she has a probability of 1 of transitioning to vertex 1. For any other vertex $i$, there is a probability of $1/2$ of transitioning to $i+1$ and a probability of $1/2$ of transitioning to $i-1$. What is the expected number of steps Alice takes to reach vertex $n$?
\Question
\textbf{Exponential LLSE}
Let $X \sim U[0, a]$ and let $Y = e^X$. Compute $L[Y \mid X]$. What does $L[Y \mid X]$ approach as $a \to 0$?
\Question
\textbf{First Exponential to Die}
Let $X$ and $Y$ be $\operatorname{Expo}(\lambda_1)$ and $\operatorname{Expo}(\lambda_2)$ respectively. What is $P(\min(X, Y) = X)$, the probability that the first of the two to die is $X$?