QUESTION 1 Suppose that you organise a game whereby a player rolls two dice and you pay the player the Rand amount of the difference in the scores. (a) Determine the probability mass function of the positive difference between the scores ob- tained from rolling the two dice. (6) (b) Calculate the mean value of X, that is, the expected difference of the scores, in this case the expected or long run average payment per game. (3) (c) What is the standard deviation of the scores, that is, calculate the standard deviation of X.(7) (d) Would you say that the distribution is symmetrical? Justify your answer. (7 ) (e) Suppose that you charge the players R2 to roll the two dice and then you pay them the difference in the scores. Let Y be the amount of money you make. (i) What are the possible values that Y can take? (6) (ii) What is your expected gain per game? (3 ) (f) Suppose you fix the dice so that each die has a probability of 0.2 of scoring a 3 and equal probability of 0.16 of scoring the other five numbers. (i) Do your expected winnings increase beyond the previous expected payment per game? Explain your answer. (6) (ii) Is this a surprise? Justify. (2) [40] N