Two dice are rolled. Let X = 1 if the dice come up doubles and let X = 0 otherwise. Let Y = 1 if the sum is 6, and let Y = 0 otherwise. Let Z = 1 if the dice come up both doubles and with a sum of 6 (that is, double 3), and let Z = 0 otherwise.

a. Let p_X denote the success probability for X. Find p_X.

b. Let p_Y denote the success probability for Y. Find p_Y.

c. Let p_Z denote the success probability for Z. Find p_Z.

d. Are X and Y independent?

e. Does p_Z = p_X p_Y?

f. Does Z = XY? Explain.