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Question 4 The sample space for the random experiment of rolling two fair dice is shown in Figure 4.1. Ly 1,2 1,3 @OL,4 1,5 0,06 2,) (2,2) (2,3) 2,4 (2,5 (2,0) G (3,2) 3,3) (G4 G5 (6 - 4,0 42 43 44 &5 4,0 G, 5,2 (53 (549 (55 (5,6 6,1) (6,2) (6,3) (6,4) (6,5 (6,6) Figure 4.1 A player in the board game SD141 rolls two fair dice, and advances his piece Z positions on the board, where Z is the square of the difference between the scores on the two dice. (For example, if the outcome of the experiment is (2, 4) or (4, 2) then Z= (4-2)* =4.) (a) Calculate the following probabilities: [9%] P(Z =0) (i1) P(Z=1) (i) P(Z=4) vy P(Z=9) v) P(Z=16) (vi) P(Z=25) (b) Use your answers to part (a) to determine the mean and variance of Z. (You should show [6%] all steps in your working.) (c) (i) Showing any formulae you use, calculate the conditional probability P(O|Z >2), [3%] where O is the event that Z is an odd number. (i) Comment on whether or not the events \"Z > 2" and O are independent, giving your [2%] reasons