Calculate E(X) from constructed distribution

The question requires first constructing or deriving the probability distribution from a scenario (dice, games, matching problems) before calculating E(X).

3 questions

CAIE S1 2016 November Q2
2 Two fair six-sided dice with faces numbered 1, 2, 3, 4, 5, 6 are thrown and the two scores are noted. The difference between the two scores is defined as follows.
  • If the scores are equal the difference is zero.
  • If the scores are not equal the difference is the larger score minus the smaller score.
Find the expectation of the difference between the two scores.
OCR MEI S1 2006 January Q2
2 Four letters are taken out of their envelopes for signing. Unfortunately they are replaced randomly, one in each envelope. The probability distribution for the number of letters, \(X\), which are now in the correct envelope is given in the following table.
\(r\)01234
\(\mathrm { P } ( X = r )\)\(\frac { 3 } { 8 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 4 }\)0\(\frac { 1 } { 24 }\)
  1. Explain why the case \(X = 3\) is impossible.
  2. Explain why \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 24 }\).
  3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
AQA Further Paper 3 Statistics 2021 June Q3
3 In a game, it is only possible to score 10, 20 or 30 points. The probability of scoring 20 points is twice the probability of scoring 30 points.
The probability of scoring 20 points is half the probability of scoring 10 points.
3
  1. Find the mean points scored when the game is played once, giving your answer to two decimal places.
    3
  2. Mina plays the game.
    Her father, Michael, tells her that he will multiply her score by 5 and then subtract 10 He will then give her the value he has calculated in pence rounded to the nearest penny. Calculate the expected value in pence that Mina receives.