Calculate expectation and variance

A question is this type if and only if it asks to calculate both E(X) and Var(X) from the probability distribution, without either being given.

3 questions

CAIE S1 2010 June Q6
6 A small farm has 5 ducks and 2 geese. Four of these birds are to be chosen at random. The random variable \(X\) represents the number of geese chosen.
  1. Draw up the probability distribution of \(X\).
  2. Show that \(\mathrm { E } ( X ) = \frac { 8 } { 7 }\) and calculate \(\operatorname { Var } ( X )\).
  3. When the farmer's dog is let loose, it chases either the ducks with probability \(\frac { 3 } { 5 }\) or the geese with probability \(\frac { 2 } { 5 }\). If the dog chases the ducks there is a probability of \(\frac { 1 } { 10 }\) that they will attack the dog. If the dog chases the geese there is a probability of \(\frac { 3 } { 4 }\) that they will attack the dog. Given that the dog is not attacked, find the probability that it was chasing the geese.
CAIE S1 2018 June Q3
3 Andy has 4 red socks and 8 black socks in his drawer. He takes 2 socks at random from his drawer.
  1. Find the probability that the socks taken are of different colours.
    The random variable \(X\) is the number of red socks taken.
  2. Draw up the probability distribution table for \(X\).
  3. Find \(\mathrm { E } ( X )\).
AQA S2 2010 January Q6
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)