AQA S2 2012 January — Question 5

Exam BoardAQA
ModuleS2 (Statistics 2)
Year2012
SessionJanuary
TopicDiscrete Probability Distributions
TypeSequential trials until success
5
  1. Joshua plays a game in which he repeatedly tosses an unbiased coin. His game concludes when he obtains either a head or 5 tails in succession. The random variable \(N\) denotes the number of tosses of his coin required to conclude a game. By completing Table 3 below, calculate \(\mathrm { E } ( N )\).
  2. Joshua's sister, Ruth, plays a separate game in which she repeatedly tosses a coin that is biased in such a way that the probability of a head in a single toss of her coin is \(\frac { 1 } { 4 }\). Her game also concludes when she obtains either a head or 5 tails in succession. The random variable \(M\) denotes the number of tosses of her coin required to conclude her game. Complete Table 4 below.
    1. Joshua and Ruth play their games simultaneously. Calculate the probability that Joshua and Ruth will conclude their games in an equal number of tosses of their coins.
    2. Joshua and Ruth play their games simultaneously on 3 occasions. Calculate the probability that, on at least 2 of these occasions, their games will be concluded in an equal number of tosses of their coins. Give your answer to three decimal places.
      (4 marks) \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Table 3}
      \(\boldsymbol { n }\)12345
      \(\mathbf { P } ( \boldsymbol { N } = \boldsymbol { n } )\)\(\frac { 1 } { 8 }\)\(\frac { 1 } { 16 }\)
      \end{table} \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Table 4}
      \(\boldsymbol { m }\)12345
      \(\mathbf { P } ( \boldsymbol { M } = \boldsymbol { m } )\)\(\frac { 1 } { 4 }\)\(\frac { 3 } { 16 }\)
      \end{table}
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