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{\Large \textbf{Sample Questions: Conditional Probability}}\footnote{STA256 Fall 2019. 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
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\item The probability of living? \vspace{10mm}
\item The probability of living
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\item Given 1st class? \vspace{20mm}
\item Given 2nd class? \vspace{20mm}
\item Given 3d class? \vspace{20mm}
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\item The probability of being in first class given that the person died? \vspace{40mm}
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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 $S = \cup_{k=1}^\infty A_k$, disjoint, with $P(A_k)>0$ for all $k$. Using the formula sheet and the tabular format illustrated in lecture, prove $P(B) = \sum_{k=1}^\infty P(B|A_k)P(A_k)$.
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\item Prove the following version of Bayes' Theorem. Let $S = \cup_{k=1}^\infty A_k$, disjoint, with $P(A_k)>0$ for all $k$. Then
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P(A_j|B) = \frac{P(B|A_j)P(A_j)}{\sum_{k=1}^\infty P(B|A_k)P(A_k)}.
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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.
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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/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}
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