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The following table summarizes the analysis of samples
of galvanized steel for coating weight and surface roughness:

$$\begin{array}{llll}{} & {} & {\frac{\text { coating weight }}{\text { high }}} & {\text { low }} \\ {\text { surface }} & {\text { high }} & {\text { 12 }} & {\text { 16 }} \\ {\text { roughness }} & {\text { low }} & {88} & {\text { 34 }}\end{array} $$

(a) If the coating weight of a sample is high, what is the prob-
ability that the surface roughness is high?
(b) If the surface roughness of a sample is high, what is the
probability that the coating weight is high?
(c) If the surface roughness of a sample is low, what is the
probability that the coating weight is low?

Coating weight high $$\rightarrow A$$

Surface finish high $$\rightarrow B$$

(a) $$P(B | A)$$

$$P(B | A)=\frac{P(A \cap B)}{P(A)} $$

$$=\frac{n(A \cap B)}{n(A)}=\frac{12}{12+88}=0.12$$

(b) $$P(A | B) \Rightarrow P(A | B)=\frac{n(A \cap B)}{n(B)}$$

$$=\frac{12}{12+16} = \frac{12}{28} $$

(c) surface roughness low $$\rightarrow C$$

coating weight low $$\rightarrow d$$

$$P(D | C)=\frac{P\left(D \cap C\right)}{P(C)} $$

$$=\frac{34}{34+88}=\frac{34}{122} \simeq 0.28$$

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