Problem 17.6. There were n Immortal Warriors born into our world, but in the end there can be only one. The Immortals’ original plan was to stalk the world for centuries, dueling one another with ancient swords in dramatic landscapes until only one survivor remained. However, after a thought-provoking discussion probability, they opt to give the following protocol a try: (i) The Immortals forge a coin that comes up heads with probability p.
Problem 17.6. There were n Immortal Warriors born into our world, but in the end there can be only one. The Immortals’ original plan was to stalk the world for centuries, dueling one another with ancient swords in dramatic landscapes until only one survivor remained. However, after a thought-provoking discussion probability, they opt to give the following protocol a try: (i) The Immortals forge a coin that comes up heads with probability p. (ii) Each Immortal flips the coin once. (iii) If exactly one Immortal flips heads, then they are declared The One. Otherwise, the protocol is declared a failure, and they all go back to hacking each other up with swords.
One of the Immortals (Kurgan from the Russian steppe) argues that as n grows large, the probability that this protocol succeeds must tend to zero. Another (McLeod from the Scottish highlands) argues that this need not be the case, provided p is chosen carefully.
(a) A natural sample space to use to model this problem is fH; T g n of length-n sequences of H and T’s, where the successive H’s and T’s in an outcome correspond to the Head or Tail flipped on each one of the n successive flips. Explain how a tree diagram approach leads to assigning a probability to each outcome that depends only on p; n and the number h of H’s in the outcome.
(b) What is the probability that the experiment succeeds as a function of p and n?
(c) How should p, the bias of the coin, be chosen in order to maximize the probability that the experiment succeeds?
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