There are three non-standard dices: Dice A has sides 2, 2, 4, 4, 9, 9; Die B has sides 1, 1, 6, 6, 8, 8; Dice C has sides 3, 3, 5, 5, 7, 7. For each dice, the probability of each side is 1/6. For example, if we roll Dice A, the probabilities of getting a 2 is 1/3.

9. [15 marks] There are three non-standard dices: Dice A has sides 2, 2, 4, 4, 9, 9; Die B has sides 1, 1, 6, 6, 8, 8; Dice C has sides 3, 3, 5, 5, 7, 7. For each dice, the probability of each side is 1/6. For example, if we roll Dice A, the probabilities of getting a 2 is 1/3. ~ 9 8 A B C (a) [4 marks] Consider the experiment of tossing Dice B. X denotes the outcome number of the experiment. What are possible values of X? What is the probability mass function of X (i.e., the probabilities that X is equal to each possible value)? Compute the mean and standard deviation of X.~ (b) [5 marks] Tom and Mike play games using the three dices. Tom chooses Dice A and Mike chooses Dice B. Tom and Mike roll their dices; the player who rolls the larger number wins. Define the following event: E = {Tom wins) . Identify the sample points included in the even E and calculate the probability of E.~ (c) [6 marks] Now, consider the following game: Step 1: Tom chooses a dice from the three non-standard dices. Step 2: Mike chooses one dice from the remaining dices. Step 3: Tom and Mike roll their dices. The player who rolls the larger number wins. Who has higher chance to win in the game? If you are Mike, which dice will you choose?<