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\begin{document}
{\bf Question}
\begin{description}
\item[(a)]
A gambler with initial capital $£z$ plays against an opponent with
initial capital $(a-z)$ where $a$ and $z$ are integers, and $0
\leq z \leq a$. At each play the gambler wins $£1$ with
probability $\df{1}{3}$,\ loses $£1$ with probability $\df{1}{2}$,
and retains his stake money in the event of a draw.
Formulate a difference equation for the probability $P_z$ that the
game ends in an even number of bets, with appropriate boundary
conditions involving $P_0$ and $P_a$. Solve the equation to find
an explicit formula for $P_z$ in terms of $z$ and $a$.
What is the value of $P_z$ if the gambler plays against an
infinitely rich opponent?
\item[(b)]
In the classical gambler's ruin problem one of the players will
eventually be ruined, with probability 1. Write a brief
explanation of what is meant by thus, giving particular attention
to the concept \lq\lq with probability 1" in this context.
\end{description}
\vspace{.25in}
{\bf Answer}
\begin{description}
\item[(a)]
Consider the first bet. There are 3 possible outcomes
\begin{description}
\item[(i)]
Gambler wins. Then he has $£z+1$ and the game must not end in a
further even number of bets. Prob: $\df{1}{3}(1-P_{z+1})$.
\item[(ii)]
Gambler loses. Then he has $£z-1$ and the game must not end in a
further even number of bets. Prob: $\df{1}{2}(1-P_{z-1})$.
\item[(iii)]
Draw. Then the gambler still has $£z$ and the game must not end in
a further even number of bets. Prob: $\df{1}{6}(1-P_z)$.
\end{description}
${}$
So
$P_z=\df{1}£{}(1-P_{z+1})+\df{1}{2}(1-P_{z-1})+\df{1}{6}(1-P_z)$
for $0