Problems

In a game for two players, a fair coin is tossed repeatedly. Each player is assigned a sequence of heads and tails and the player whose sequence appears first wins. Four players, \(A\), \(B\), \(C\) and \(D\) take turns to play the game. Each time they play, \(A\) is assigned the sequence TTH (i.e.~Tail then Tail then Head), \(B\) is assigned THH, \(C\) is assigned HHT and \(D\) is assigned~HTT.

1. \(A\) and \(B\) play the game. Let \(p_{\mathstrut\mbox{\tiny HH}}\), \(p_{\mathstrut\mbox{\tiny HT}}\), \(p_{\mathstrut\mbox{\tiny TH}}\) and \(p_{\mathstrut\mbox{\tiny TT}}\) be the probabilities of \(A\) winning the game given that the first two tosses of the coin show HH, HT, TH and TT, respectively. Explain why \(p_{\mathstrut\mbox{\tiny TT}} = 1\,\), and why $p_{\mathstrut\mbox{\tiny HT}} = {1 \over 2} \, p_{\mathstrut\mbox{\tiny TH}} + {1\over 2} \, p_{\mathstrut\mbox{\tiny TT}}\,$. Show that $p_{\mathstrut\mbox{\tiny HH}} = p_{\mathstrut\mbox{\tiny HT}} = {2 \over 3}$ and that \(p_{\mathstrut\mbox{\tiny TH}} = {1\over 3}\,\). Deduce that the probability that A wins the game is \({2\over 3}\,\).
2. \(B\) and \(C\) play the game. Find the probability that \(B\) wins.
3. Show that if \(C\) plays \(D\), then \(C\) is more likely to win than \(D\), but that if \(D\) plays \(A\), then \(D\) is more likely to win than \(A\).

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?