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Bread roll throwing duels at the Drones' Club are governed by a strict
etiquette. The two duellists throw alternatively until one is hit,
when the other is declared the winner. If Percy has probability $p>0$
of hitting his target and Rodney has probability $r>0$ of hitting
his, show that, if Percy throws first, the probability that he beats
Rodney is 
\[
\frac{p}{p+r-pr}.
\]
Algernon, Bertie and Cuthbert decide to have a three sided duel in
which they throw in order $\mathrm{A,B,C,A,B,C,}\ldots$ except that
anyone who is hit must leave the game. Cuthbert always his target,
Bertie hits his target with probability $3/5$ and Algernon hits his
target with probability $2/5.$ Bertie and Cuthbert will always aim
at each other if they are both still in the duel. Otherwise they aim
at Algernon. With his first shot Algernon may aim at either Bertie
or Cuthbert or deliberately miss both. Faced with only one opponent
Algernon will aim at him. What are Algernon's changes of winning if
he: 
\begin{itemize}
\setlength{\itemsep}{3mm}
\item[\bf (i)] hits Cuthbert with his first shot?
\item[\bf (ii)] hits Bertie with his first shot?
\item[\bf (iii)] misses with his first shot?
\end{itemize}
Advise Algernon as to his best plan and show that, if he uses this
plan, his probability of winning is $226/475.$

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