\Question{Identity Theft}
A group of $n$ friends go to the gym together, and while they are playing basketball, they leave their bags against the nearby wall.
An evildoer comes, takes the student ID cards from the bags, randomly rearranges them, and places them back in the bags, one ID card per bag.
What is the probability that no one receives his or her own ID card back?
[\textit{Hint}: Use the generalized inclusion-exclusion principle.]
Then, find an approximation for the probability as $n \to \infty$. You may, without proof, refer to the power series for $\e^x$: $$\e^x = \sum_0^{\infty} \frac{x^k}{k!}$$
\Question{Mario's Coins}
Mario owns three identical-looking coins. One coin shows heads with probability $1/4$, another shows heads with probability $1/2$, and the last shows heads with probability $3/4$.
\begin{Parts}
\Part Mario randomly picks a coin and flips it. He then picks one of the other two coins and flips it. Let $X_1$ and $X_2$ be indicators of the 1st and 2nd flips showing heads. Are $X_1$ and $X_2$ independent? If so, prove it; if not, provide a counterexample.
\Part Mario randomly picks a single coin and flips it twice. Let $Y_1$ and $Y_2$ be indicators of the 1st and 2nd flips showing heads. Are $Y_1$ and $Y_2$ independent? If so, prove it; if not, provide a counterexample.
\Part Mario arranges his three coins in a row. He flips the coin on the left, which shows heads. He then flips the coin in the middle, which shows heads. Finally, he flips the coin on the right. What is the probability that it also shows heads?
\end{Parts}
\Question{Combinatorial Coins}
Allen and Alvin are flipping coins for fun. Allen flips a fair coin $k$ times and Alvin flips $n-k$ times. In total there are $n$ coin flips.
\begin{Parts}
\Part Use a combinatorial proof to show that $$\sum_{i=0}^k \binom{k}{k - i} \binom{n - k}{i} = \binom{n}{k}.$$
You may assume that $n - k \geq k$.
\Part Prove that the probability that Allen and Alvin flip the same number of heads is equal to the probability that there are a total of $k$ heads.
\end{Parts}
\Question{Sinho's Dice}
Sinho rolls three fair-sided dice. What is the PMF for the maximum of the three values rolled? [\textit{Hint}: First find the CDF.]