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{\Large \textbf{Sample Questions: Conditional Probability}}\footnote{STA256 Fall 2018. Copyright information is at the end of the last page.}
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\begin{enumerate} 

\item The table below shows percentages of passengers on the Titanic.
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\begin{tabular}{l | c c}  
          & Died & Lived \\ \hline
1st Class & ~9   & 15    \\
2nd Class & 13   & ~9    \\
3d~ Class & 40   & 14    
\end{tabular}
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For a randomly chosen passenger, what is 
        \begin{enumerate}
            \item The probability of living? \vspace{10mm}
            \item The probability of living
            \begin{enumerate}
                \item Given 1st class? \vspace{20mm}
                \item Given 2nd class? \vspace{20mm}
                \item Given 3d class? \vspace{20mm}
            \end{enumerate}

            \item The probability of being in first class given that the person died? \vspace{40mm}
        \end{enumerate}
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\item  {\large A jar contains two fair coins and one fair die. The coins have a ``1" on one side and a ``2" on the other side. Pick an object at random, roll or toss, and observe the number.}
    \begin{enumerate}
        \item What is $P(2 \cap C)$? \vspace{20mm}
        \item What is $P(6|C)$? \vspace{10mm}
        \item Make a tree diagram. \vspace{90mm}
        \item List the outcomes with their probabilities. \vspace{40mm}
        \item What is $P(C|2)$?
    \end{enumerate}


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\item %{\large 
Let $\Omega = \cup_{k=1}^\infty B_k$, disjoint, with $P(B_k)>0$ for all $k$. Using the formula sheet and the tabular format illustrated in lecture, prove $P(A) = \sum_{k=1}^\infty P(A|B_k)P(B_k)$.
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\item On the Titanic, 62.5\% of the first-class passengers survived, 40.9\% of the second class passengers survived, and 25.9\% of the third class passengers survived. If 24\% of the passengers were first class, 22\% were second class and 54\% were third class, what percent of the passengers survived overall? % 0.38

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\item  Prove the following version of Bayes' Theorem. Let $\Omega = \cup_{k=1}^\infty B_k$, disjoint, with $P(B_k)>0$ for all $k$. Then 
\begin{displaymath}
    P(B_j|A) = \frac{P(A|B_j)P(B_j)}{\sum_{k=1}^\infty P(A|B_k)P(B_k)}.
\end{displaymath}
You may use anything from the formula sheet except Bayes' theorem itself.
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\item  Two balls are drawn in succession from a jar containing three red balls and four white balls. What is the probability that the first ball was white given that the second ball was red? % 2/3
The answer is a number. Circle your answer.
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\item This is an important real-world application of Bayes' Theorem. Suppose only one person in a thousand has some rare disease. We have a screening test for the disease, and it's a good test.
    \begin{itemize}
        \item 90\% of those with the disease test positive.
        \item 95\% of those without the disease test negative.
    \end{itemize}
Given a positive test, what is the probability that the person actually has the disease? 
The answer is a number. Circle your answer.

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This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website: 
     
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\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f18} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f18}}
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