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{\Large \textbf{Sample Questions: Foundations of Probability}}%\footnote{}
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STA256 Fall 2019. Copyright information is at the end of the last page.
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\item Prove Property 5: $P(A^c) = 1-P(A)$. Use Properties 1-4 of probability and the tabular format illustrated in lecture.
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\item  Prove Property 6:  If $A \subseteq B$ then $P(A) \leq P(B)$. Use Properties 1-4 of probability and the tabular format illustrated in lecture.
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\item  Prove Property 7 (the inclusion-exclusion principle): $P(A \cup B) = P(A)+P(B)-P(A\cap B)$. Use Properties 1-4 of probability and the tabular format illustrated in lecture.
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\item If 23 out of 25 are employed, what is the probability of randomly choosing an unemployed person?  The answer is a number. Circle your answer.
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\item If you roll two fair dice, what is the probability of getting a sum greater than 2? The answer is a number. Circle your answer.
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\item If you roll two fair dice, what is the probability of getting two different numbers? Your answer is a number. Circle your answer.
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\item $P(A)=0.4$, $P(B)=0.5$ and $P(A\cap B)=0.3$. What is $P(A\cup B)$? The answer is a number. Circle your answer. 
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\item Of the cars in a used car lot, 50\% have engine trouble and 50\% have transmission trouble. If 25\% have both problems and you buy a car at random, what is the probability that both the engine and transmission are okay? The answer is a number. Circle your answer. 
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\item  Let $A_1, A_2, \ldots$ form a partition of the sample space $S$, meaning that $A_1, A_2, \ldots$ are disjoint and $S = \cup_{k=1}^\infty A_k$. Let $B$ be any event. Show that $P(B) = \sum_{k=1}^\infty (A_k \cap B)$. Use the 4 basic properties of probability and the tabular format illustrated in lecture.
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\item  Let $A_1, A_2, \ldots$ be a collection of events, not necessarily disjoint. Show that
    $P\left( \cup_{k=1}^\infty A_k\right) \leq \sum_{k=1}^\infty P(A_k)$.  Use the Properties 1-7 of probability and the tabular format illustrated in lecture.
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% I might have to can these.

\item  Let $A_1 \subseteq A_2 \subseteq A_3 \subseteq \ldots$ and let $A = \cup_{k=1}^\infty A_k$. Show that $\lim_{k \rightarrow \infty} P(A_k) = P(A)$.  Use Properties 1-4 of probability and the tabular format illustrated in lecture.
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\item  Let $A_1 \supseteq A_2 \supseteq A_3 \supseteq \ldots$ and let $A = \cap_{k=1}^\infty A_k$. Show that $\lim_{k \rightarrow \infty} P(A_k) = P(A)$.  Use Properties 1-4 of probability and the tabular format illustrated in lecture.

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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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\item Of the prisoners in a jail, 75\% are convited murderers and 50\% have been convicted of both murder and armed robbery. Twenty percent are in jail for offences other than murder or armed robbery. If you pick a prisoner at random, what is the probability that she is an armed robber?
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