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\begin{center}
\Huge \textbf{Exercise 1}
\end{center}
\vspace{1cm}

\begin{enumerate}

\item An urn contains five balls numbered 1 to 5 of which the first three are
silver and the last two are gold.   A sample of size two is drawn
with replacement.   Let $E_1$ denote the event that the first ball
drawn is silver, and $E_2$ denote the event that the second ball
drawn is silver.
\begin{enumerate}
   \item \label{a} Describe the sample space for the experiment, and sketch the events $E_1$,
$E_2$ and $E_1\cap E_2$

   \item \label{b} Find $P(E_1)$, $P(E_2)$ and $P(E_1\cap E_2).$

   \item Repeat (a)  and (b) for sampling without replacement.
  \end{enumerate}

\item A garage has twelve motors to sell, two of which are faulty.
The seller can crate the motors with all twelve in one box or with
six in two boxes. The customer will inspect two of the twelve
motors if they are crated in one box, and one motor from each of
the two boxes if they are crated six to a box. The garage has
three possible strategies:
\begin{enumerate}
\item Crate all twelve in one box;
\item Put one faulty motor in each of two boxes of six.
\item Put the two faulty motors in one of the two boxes of six.
\end{enumerate}
What is the probability that the customer will not inspect either of the faulty
motors under each of the three strategies?

\item The game of ``odd one out" is played by three people, each
flipping a single coin. All three flip their coins simultaneously;
if one face is different from the other two, its owner loses the
game.
\begin{enumerate}
\item What is the probability that there is a loser on a given turn,
assuming all coins are fair?
\item If there is no loser on the first turn, the coins are all flipped again
until there is a loser. What is the probability that an even
number of turns is required to determine who loses the game?
\end{enumerate}

\item Person $A$ spins an fair coin three times and
$B$ spins it twice. By considering all possible results, show that
$A$ has probability $\frac{1}{2}$ of getting more heads than B.
Show that the same is true if A spins the coin n+1 times and B n
times.

\item If 6 people, among whom are $A$ and $B$, stand in a row, what is the
probability that there will be exactly $r$ people between $A$ and
$B$?  If they stand in a ring, show that the probability of
exactly $r$ people between $A$ and $B$ in the clockwise direction
is $1/5$. Try replacing 6 by $n$ for general problems of the same
sort.

\item An urn contains 2 red disks lettered A and B, 2 green disks
lettered A and B, and 4 blue disks lettered A, B, C and D. One disk is
drawn and its colour and letter noted. Are the following events
independent? \newline
\begin{tabular}{lllll}
 A, red; & & (A or D), red; & & (A or blue), red.
\end{tabular}

\newpage

\item  A fair coin is independently tossed twice. Consider events:
\begin{itemize}
\item $A=\{\text{The first toss is heads}\}$
\item $B=\{\text{The second toss is heads}\}$
\item $C=\{\text{First and second toss show the same side}\}$.
\end{itemize}
Show that $A,B,C$ are pairwise independent events, but not
independent events.


\item Suppose that the genders of all children in a family are independent and
that boys and girls are equally probable, that is, both have
probability 0.5.
\begin{enumerate}
\item  For families of three children calculate the probabilities of the events $A$,
$B$ and $A\cap B$ where $A =$ `There are children of both
genders', $B =$ `Not more than one child is a girl'.
\item  Do the same for families of four children.
\end{enumerate}
Are $A$ and $B$ independent events in 2(a) and 2(b)?


\item $A$, $B$, and $C$ are events of positive probability.   For each of the
following either supply a proof or a counter-example.
\begin{enumerate}
\item If $A$ and $B$ are independent, $A$ and $B^c$ are independent.
\item If $P(A \mid B) < P(A)$, then $P(B \mid A) < P(B).$
\item If $P(A\cap B \mid C) = P(A \mid C)P(B \mid C)$ then $A$ and $B$ are
independent.
\item If $P(B \mid A) = P(B \mid A^c)$ then $A$ and $B$ are independent.
\end{enumerate}

\item An fair die is thrown n times. Find a formula for the
probability that each face appears at least once. [Hint: let $A_i$
be the event that the $i$th face does not appear and expand $P(A_1
\cup \ldots \cup A_6)$.]

\item Let $X$ and $Y$ be mutually exclusive events with known
probabilities $P(X)$ and $P(Y)$. Give an expression for $P(X \cup Y)$.
By partitioning appropriate sets into disjoint subsets and using just
the result for $P(X \cup Y)$, carefully prove;
\begin{enumerate}
\item $P(A^c) = 1 - P(A)$,
\item $P(A^c \cap B) = P(B) - P(A \cap B)$,
\item $P(A \cup B) = P(A) + P(B) - P(A \cap B)$,
\item $P((A \cup B) \cap (A^c \cup B^c)) = P(A) + P(B) - 2 P(A \cap B)$.
\end{enumerate}
How would you interpret the event $(A \cup B) \cap (A^c \cup
B^c)$?

\item In a colony of
exceedingly rare sea worms, the individuals are either red or
green.
\begin{enumerate}
\item Among the population, 10\% of the worms are vicious.
One fifth of
vicious worms are green and one quarter of non-vicious worms are
red. While diving we see a red worm; what is the probability that
it is vicious?
\item A worm's colour is partly determined by the colour of its
ancestors (the worms reproduce by budding). The conditional
probabilities are $\frac{3}{4}$, $\frac{2}{3}$ and $\frac{4}{5}$
conditioning respectively on the parent being red, the grandparent
being red, and both the parent and grandparent being red. A worm
is chosen at random.
\begin{enumerate}
\item What is the probability that the worm is red
and has a red parent?
\item What is the probability that the worm is red or its parent
is red or its grandparent is red?
\end{enumerate}
\end{enumerate}

\item One urn contains four red disks and five green, and a second
contains five red and four green. One disk is transferred from the
first to the second urn and a disk is then withdrawn from the second
urn. Find the probability that it is red. What is the probability of a
red disk if two disks are transferred rather than one?



\end{enumerate}
