4. A red and a black die are thrown. Let X be the sum of the dice. (a) What is the probability of X being at least 10? [2 marks] (b) What is the conditional probability of X being at least ten, given that the black die shows five? y3 [2 marks] (c) What is the conditional probability of X being at least ten, given that at least one of the dice shows five? 341 [3 marks] (d) Let Y be the difference (red minus black). Determine the probability distributions of X and Y. [5 marks] (e) Are X and Y independent? [2 marks] (f) What are the expected value and standard deviation of X? [4 marks] 7; 2.42 (g) What is the probability that X lies within one standard deviation of the expected value? 2/3 [4 marks] [Total: 22 marks]