An unfair coin has probability 0 < p < 1 of landing heads. As an experiment,...

  

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An unfair coin has probability 0 < p < 1 of landing heads. As an experiment, the coin is thrown repeatedly, until the first time it lands tails, after which the experiment ends. Suppose that this full experiment is conducted n times in total, dur- ing which the total number of coin throws is N (this includes all n experiments). (a) Invent any suitable value for p to apply for the entire question. Name a probability distribution that is related to the number of throws in a single experiment. (b) Explain in words why we can expect that, on average, N(1 - p) = n. (8 marks) (c) Use part (b) to explain why the expected number of throws in a single experiment is 1/(1 - p). (5 marks) (d) Now suppose that the experiment is modified so that it stops only after t coins have landed tails, and before each throw, the thrown coin is selected randomly between two coins (each with probability 0.5) : one coin has probability of landing heads and the other coin has probability of landing heads. Choose t to be any integer value greater than 20 . Find the expected total number of throws for the full modified experiment to be performed 3 times in a row. An unfair coin has probability 0 < p < 1 of landing heads. As an experiment, the coin is thrown repeatedly, until the first time it lands tails, after which the experiment ends. Suppose that this full experiment is conducted n times in total, dur- ing which the total number of coin throws is N (this includes all n experiments). (a) Invent any suitable value for p to apply for the entire question. Name a probability distribution that is related to the number of throws in a single experiment. (b) Explain in words why we can expect that, on average, N(1 - p) = n. (8 marks) (c) Use part (b) to explain why the expected number of throws in a single experiment is 1/(1 - p). (5 marks) (d) Now suppose that the experiment is modified so that it stops only after t coins have landed tails, and before each throw, the thrown coin is selected randomly between two coins (each with probability 0.5) : one coin has probability of landing heads and the other coin has probability of landing heads. Choose t to be any integer value greater than 20 . Find the expected total number of throws for the full modified experiment to be performed 3 times in a row.