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. Is it possible for both X and Y to equal 1?
e. Does 
?
f. Does Z = X + Y? Explain.
(a) If X and Y cannot both be equal to 1, the possible values for Z are 0 and 1, so Z is a Bernoulli random variable.
(b) 
= P(Z = 1) = P(X = 1 or Y = 1) = P(X = 1) + P(Y = 1) - P(X = 1 and Y = 1) = P (X = 1) + P (Y = 1)-0 =
(c) P(Z=2) = P(X=1 and Y = 1). Since P(z=2) ? 0, Z is not a Bernoulli random variable.
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