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\section{Probability Theory 1}

\textbf{Problem 1:} You walk into Ye Olde Dice Tavern in South Loop to play a game of dice against a random stranger. You know that only one in a thousand dice players use loaded dice, which have a 90\% chance of coming up $6$ (everyone else uses normal dice). Lo and behold, your opponent's first four rolls all come up heads. What is the probability that she is using loaded dice?\\[10pt]

\textbf{Problem 2:} Suppose you roll $2$ dice. 
Are the following events positively correlated, negatively correlated or independent? Show your work.
\begin{enumerate}
    \item The number on the first die is odd and the total is even.
    \item The first die shows a $6$ and the absolute difference between the dice is $1$.
    \item The absolute difference between the dice is $2$ and the total is $10$.
    \item The second die shows a $3$ and the difference between the two dice is $6$.
    \item The first die shows a $4$ and the total is $10$ or greater.
\end{enumerate}~\\[10pt]


\textbf{Problem 3:} In a class of 58 people, all of whom were born in the year 2000, what is the probability that exactly 2 were born on Jan 1st, 2000? (Note: The year 2000 was a leap year.)\\[10pt]

\textbf{Problem 4:} In a simplified game of Blackjack, we assign values to cards as follows: For each number card $i$, $V(i) = i$, and for face cards  $V(J)=V(Q)=V(K) = 10$, and $V(A) = 11$. What is the expected value of a hand of two cards?
 


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