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Confidence of feedback information for improving quality. In the semiconductor manufacturing industry, a key to improved quality is having confidence in the feedback generated by production equipment. A study of the confidence level of feedback information was published in Engineering Applications of Artificial Intelligence(Vol. 26, 2013). At any point in time during the production process, a report can be generated. The report is classified as either 鈥淥K鈥� or 鈥渘ot OK.鈥� Let Arepresent the event that an 鈥淥K鈥� report is generated in any time period (t).Let Brepresent the event that an 鈥淥K鈥� report is generated in the next time period. Consider the following probabilities:

$\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{8}$,$\mathbf{P}\left(\frac{B}{A}\right)\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{9}$, and$\mathbf{P}\left(\frac{B}{{\mathbf{A}}^C}\right)\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5}$.

a. Express the event B|Ain the words of the problem.

b. Express the event B|${\mathbf{A}}^C$in the words of the problem.

c. Find$\mathbf{P}\mathbf{(}{\mathbf{A}}^C\mathbf{)}$.

d. Find$\mathbf{P}\mathbf{(}\mathbf{A}鈭�\mathbf{B}\mathbf{)}$.

e. Find$\mathbf{P}\mathbf{(}{\mathbf{A}}^C鈭�\mathbf{B}\mathbf{)}$.

f. Use the probabilities, parts d and e, to find P(B).

g. Use Bayes鈥� Rule to find P(A|B), i.e., the probability that an 鈥淥K鈥� report was generated in one time period(t), given that an 鈥淥K鈥� report is generated in the next time period$\left(t+1\right)$.

Step by step solution

01

Given information

$\begin{array}{c}\mathrm{P}\left(\mathrm{A}\right)=0.8\\ \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)=0.9\\ \mathrm{P}\left(\frac{\mathrm{B}}{{\mathrm{A}}^{\mathrm{C}}}\right)=0.5\end{array}$

Let A represent the event that an 鈥淥K鈥� report is generated in any time period (t).Let B represent the event that an 鈥淥K鈥� report is generated in the next time period .

02

Express the event B|A in the words of the problem.

A=[in the time period (t) an OK report id generated]

B=[ in the time period$\left(t+1\right)$ an OK report id generated]

So,If an OK report is generated during time period $\left(t+1\right)$, then the event B|A indicates that it was done so (t).

03

Express the event B|AC in the words of the problem

Thereafter, unless a not-OK report is made during the time period $\left(t+1\right)$, the event B| denotes that an OK report was generated during that time period (t).

04

Find P(AC)

Apply the complementary rule

$\begin{array}{c}\mathrm{P}\text{(}{\mathrm{A}}^{\mathrm{C}}\text{)}=1-\mathrm{P}\text{(}\mathrm{A}\text{)}\\ =1-0.8\\ =0.2\end{array}$

Accordingly, the value of a compliment is 0.2

05

Evaluate P(A∩B)

$\begin{array}{c}\mathrm{P}\text{(}\mathrm{A}鈭�\mathrm{B}\text{)}=\mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)脳\mathrm{P}\text{(}\mathrm{A}\text{)}\\ =\text{(}0.9\text{)(}0.8\text{)}\\ =0.72\end{array}$

Henceforth, the value of$\mathrm{P}\text{(}\mathrm{A}鈭�\mathrm{B}\text{)}$ is 0.72.

06

Determine P(AC∩B)

$\begin{array}{c}\mathrm{P}\text{(}{\mathrm{A}}^{\mathrm{C}}鈭�\mathrm{B}\text{)}=\mathrm{P}\left(\frac{\mathrm{B}}{{\mathrm{A}}^{\mathrm{C}}}\right)脳\mathrm{P}\text{(}{\mathrm{A}}^{\mathrm{c}}\text{)}\\ =\text{(}0.5\text{)(}0.2\text{)}\\ =0.1\end{array}$

So, the value of$\mathrm{P}\text{(}{\mathrm{A}}^{\mathrm{C}}鈭�\mathrm{B}\text{)}$ is 0.1.

07

Find P(B)

$\begin{array}{c}\mathrm{P}\text{(}\mathrm{B}\text{)}=\mathrm{P}\text{(}\mathrm{A}鈭�\mathrm{B}\text{)}+\mathrm{P}\text{(}{\mathrm{A}}^{\mathrm{C}}鈭�\mathrm{B}\text{)}\\ =0.72+0.1\\ =0.82\end{array}$

Hence, the value of (B) is 0.82.

08

Find P(A|B)

$\begin{array}{c}\mathrm{P}\left(\mathrm{A}|\mathrm{B}\right)=\frac{\mathrm{P}\text{(}\mathrm{A}鈭�\mathrm{B}\text{)}}{\mathrm{P}\text{(}\mathrm{A}\text{)}}\\ =\frac{\mathrm{P}\text{(}\mathrm{A}\text{)}.\mathrm{P}\text{(}\mathrm{B}\text{)}}{\mathrm{P}\text{(}\mathrm{A}\text{)}}\\ =0.82\end{array}$

Therefore, the value of P|B is 0.82.

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