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\textbf{Probability and Statistics (SPAS230C)\\ Problem Set I}
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\begin{enumerate}
\item Let $E, F, G$ be three events. Find expressions for the events that of $E, F, G$
\begin{enumerate}
\item only $F$ occurs
\item both $E$ and $F$ but not $G$ occur
\item at least one event occurs
\item at least two event occur
\item all three events occur
\item none occurs
\item at most one event occurs
\item at most two events occur
\end{enumerate}
\noindent \textbf{Answer:}
\textbf{(a)} $F\cap E^{c}\cap G^{c}$, \textbf{(b)} $E\cap F\cap G^{c}$, \textbf{(c)} $E\cup F\cup G$,
\textbf{(d)} $(E\cap F)\cup (E\cap G)\cup (F\cap G)$, \textbf{(e)} $E\cap F\cap G$,
\textbf{(f)} $(E\cup F\cup G)^c$, \textbf{(g)} $(E\cap F)^c\cap (E\cap G)^c\cap (F\cap G)^c$,
\textbf{(h)} $(E\cap F\cap G)^c$

\item Let $S = \{0, 1, 2, \cdots\}$ and $E \subseteq S$. Then in each of the following cases, verify $P$ is a probability on $S$.
\begin{enumerate}
\item $P(E) = \displaystyle {\sum_{x \in E}}\frac{e^{-\lambda}\lambda^{x}}{!x}$, $\lambda > 0$.
\item $P(E) = \displaystyle {\sum_{x \in E}}p (1-p)^x$, $0 < p < 1$.
\item $P(E) = 0$, if $E$ has finite number of elements, and $P(E) = 1$, if $E$ has infinite number of elements.
\end{enumerate}
\noindent \textbf{Answer:} (a) Yes (b) Yes (c) No
\item For events $E_1, E_2, \cdots, E_n$, show that 
\begin{enumerate}
\item $P(\displaystyle{\bigcup_{i = 1}^{n}E_i}) \leq \displaystyle {\sum_{i = 1}^{n}}P(E_i)$.\\
\noindent This is known as Boole's inequality.
\item $P(\displaystyle{\bigcap_{i = 1}^{n}E_i}) \geq \displaystyle {\sum_{i = 1}^{n}}P(E_i) - (n -1)$.\\
\noindent This is known as Bonferroni's inequality.
\item $P(\displaystyle{\bigcap_{i = 1}^{n}E_i}) = P(E_1)P(E_2|E_1)P(E_3|E_1\cap E_2)\cdots P(E_n|E_1\cap E_2\cap \cdots \cap E_{n-1})$.
\end{enumerate}
\item Let $E$ and $F$ be two independent events. Then show that
\begin{enumerate}
\item $E^c$ and $F$ are independent.
\item $E$ and $F^c$ are independent.
\item $E^c$ and $F^c$ are independent.
\end{enumerate}
 \item Let $E$ and $F$ be two events such that $P(E) = p_1 > 0$, $P(F) = p_2 > 0$ and $p_1 + p_2 >1$. Show that $P(F|E) \geq 1- \frac{1-p_2}{p_1}$.
 \item For any two events $E$ and $F$, show that $P(E\cap F) - P(E)P(F) = P(E)P(F^c) -P(E\cap F^c)= P(E^c)P(F) -P(E^c\cap F) = P((E\cup F)^c) - P(E^c)P(F^c)$.
 \item Two digits are chosen at random without replacement from the set of integers $\{1, 2, 3, 4, 5,$ $6, 7, 8\}$.
\begin{enumerate}
\item Find the probability that both digits are greater than $5$.
\item Show that the probability that the sum of the digits will be equal to $5$ is the same as the probability that their sum will exceed $13$.
\end{enumerate}
\noindent \textbf{Answer:} (a) $\frac{{\displaystyle {3 \choose 2}}}{{\displaystyle {8 \choose 2}}}$
\item Consider two independent fair coins tosses, in which all four possible outcomes are equally likely. Let $H_1 = \{$1st toss is a head$\}$, $H_2 = \{$2nd toss is a head$\}$, and $D$ = $\{$the two tosses have different results$\}$. Find $P(H_1), P(H_2), P(H_1\cap H_2), P(H_1|D), P(H_2|D)$, and $P(H_1\cap H_2|D)$.

\noindent \textbf{Answer:} $P(H_1) = \frac{1}{2} = P(H_2), P(H_1\cap H_2) = \frac{1}{4}, P(H_1|D) = \frac{1}{2} = P(H_2|D)$, and $P(H_1\cap H_2|D) = 0$.
\item Suppose that we have $n \geq 2$ letters and corresponding $n$ addressed envelopes. If these letters are inserted at random in $n$ envelopes, find the probability that no letter is inserted into the correct envelope.

\noindent \textbf{Answer:} $\frac{1}{2!}-\frac{1}{3!}+ \cdots + (-1)^{n+2}\frac{1}{n!}$.
\item  What is the chance that a leap year, selected at random, will contain 53 Sundays?

\noindent \textbf{Answer:} $\frac{2}{7}$.
\item Three numbers are chosen from 1 to 20. Find the probability that they are not consecutive.

\noindent \textbf{Answer:} $\frac{187}{190}$
\item Three numbers are chosen at random and without replacement from the set $\{1, 2, . . . , 50\}$. Find the probability that the chosen numbers are in (a) arithmetic progression, and (b) geometric progression.

\noindent \textbf{Answer:} (a) $\frac{600}{{\displaystyle {50 \choose 3}}}$ (b) (a) $\frac{44}{{\displaystyle {50 \choose 3}}}$
\item A family has two children. What is the conditional probability that both are boys given that at least one of them is a boy?

\noindent \textbf{Answer:} $\frac{1}{3}$
 \item The organization that David Jones works for is running a father son dinner for those employees having at least one son. Each of these employees is invited to attend along with his youngest son. If Jones is known to have two children, what is the conditional probability that they are both boys given that he is invited to the dinner?

\noindent \textbf{Answer:} $\frac{1}{3}$
\item An urn contains $b$ black balls and $r$ red balls. One of the balls is drawn at random, but when it is put back in the urn $c$ additional balls of the same color are put in with it. Now suppose that we draw another ball. Find the probability that the first ball drawn was black given that the second ball drawn was red.

\noindent \textbf{Answer:} $\frac{b}{b+r+c}$
\end{enumerate}
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