AQA S2 2010 January — Question 6

Exam BoardAQA
ModuleS2 (Statistics 2)
Year2010
SessionJanuary
TopicHypergeometric Distribution
TypeCalculate expectation and variance
6
  1. Ali has a bag of 10 balls, of which 5 are red and 5 are blue. He asks Ben to select 5 of these balls from the bag at random. The probability distribution of \(X\), the number of red balls that Ben selects, is given in Table 1. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Table 1}
    \(\boldsymbol { x }\)012345
    \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)\(\frac { 1 } { 252 }\)\(\frac { 25 } { 252 }\)\(\frac { 100 } { 252 }\)\(a\)\(\frac { 25 } { 252 }\)\(\frac { 1 } { 252 }\)
    \end{table}
    1. State the value of \(a\).
    2. Hence write down the value of \(\mathrm { E } ( X )\).
    3. Determine the standard deviation of \(X\).
  2. Ali decides to play a game with Joanne using the same 10 balls. Joanne is asked to select 2 balls from the bag at random. Ali agrees to pay Joanne 90 p if the two balls that she selects are the same colour, but nothing if they are different colours. Joanne pays 50 p to play the game. The probability distribution of \(Y\), the number of red balls that Joanne selects, is given in Table 2. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Table 2}
    \(\boldsymbol { y }\)012
    \(\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )\)\(\frac { 2 } { 9 }\)\(\frac { 5 } { 9 }\)\(\frac { 2 } { 9 }\)
    \end{table}
    1. Determine whether Joanne can expect to make a profit or a loss from playing the game once.
    2. Hence calculate the expected size of this profit or loss after Joanne has played the game 100 times.
      (3 marks)
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