When a certain glaze is applied to a ceramic surface, the probability is 5% that there will be discoloration, 20% that there will be a crack, and 23% that there will be either discoloration or a crack, or both. Let X = 1 if there is discoloration, and let X = 0 otherwise. Let Y = 1 if there is a crack, and let Y = 0 otherwise. Let Z = 1 if there is either discoloration or a crack, or both, and let Z = 0 otherwise.
a. Let 
denote the success probability for X. Find .
b. Let 
denote the success probability for Y. Find .
c. Let 
denote the success probability for Z. Find .
d. Are X and Y independent?
e. Does 
f. Does Z = XY? Explain.
a) = 0.05
(b) = 0.20
(c) = 0.23
(d) Yes, it is possible for there to be both discoloration and a crack.
(e) No. 
(f)No. If the surface has both discoloration and a crack, then ????= 1, ????= 1, and ????= 1, but ????+ ????= 2.
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