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{\Large \textbf{Sample Questions: Independence}}

STA256 Fall 2019. Copyright information is at the end of the last page.
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\item A jar contains 5 red balls and 15 black balls. Draw 2 balls randomly with replacement. 
    \begin{enumerate}
        \item What is the probability that the first ball is red and the second is black? The answer is a number. \vspace{85mm}
        \item What is the probability of one red and one black in any order? The answer is a number. 
    \end{enumerate}

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\item Let $A$ and $B$ be events of positive probability. Is it possible for $A$ and $B$ to be both disjoint and independent? Answer Yes or No and prove your answer.

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\item Roll a fair die $n$ times. 
    \begin{enumerate}
        \item What is the probability of observing at least one 4?  \vspace{85mm}
        \item How many times must you roll the die for the probability of at least one 4 to be 0.90 or more? The answer is a number. 
    \end{enumerate}

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\item A biased coin has $P(\mbox{Head}) = p$. Toss it three times.
    \begin{enumerate}
        \item List the elements of the sample space, along with their probabilities.  \vspace{70mm}
        \item What is $P(\mbox{Two Heads})$?  \vspace{20mm}
    \end{enumerate}

\item It is clear from the last problem that the probability of a string with $k$ heads is the same, regardless of their placement. Suppose we toss the biased coin $n$ times. What is the probability of $k$ heads (for $k = 0, \ldots, n$)? 

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\item Again, a biased coin has $P(\mbox{Head}) = p$. Toss it until the first head occurs, and then stop. 
    \begin{enumerate}
        \item What is the probability that the first head appears on the fifth toss? \vspace{30mm}
        \item What is the probability that a head eventually occurs (on toss 1 or 2 or \ldots)? \pagebreak
        \item  What is the probability that the first head occurs on an even numbered toss \linebreak
        (toss 2 or 4 or \ldots)?
    \end{enumerate}

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\item In repeated tosses of a coin with $P(\mbox{Head})=\theta$, what is the probability that the third head comes on the seventh toss?

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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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