Problems

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:

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• \(\bf (i)\) hits Cuthbert with his first shot?
• \(\bf (ii)\) hits Bertie with his first shot?
• \(\bf (iii)\) misses with his first shot?