Answer and explanation:
Given : The probabilities of poor print quality given no printer problem, misaligned paper, high ink viscosity, or printer-head debris are 0, 0.3, 0.4, and 0.6, respectively.
The probabilities of no printer problem, misaligned paper, high ink viscosity, or printer-head debris are 0.8, 0.02, 0.08, and 0.1, respectively.
Let the event E denote the poor print quality.
Let the event A be the no printer problem i.e. P(A)=0.8
Let the event B be the misaligned paper i.e. P(B)=0.02
Let the event C be the high ink viscosity i.e. P(C)=0.08
Let the event D be the printer-head debris i.e. P(D)=0.1
and the probabilities of poor print quality given printers are
![P(E|A)=0,\ P(E|B)=0.3,\ P(E|C)=0.4,\ P(E|D)=0.6](https://tex.z-dn.net/?f=P%28E%7CA%29%3D0%2C%5C%20P%28E%7CB%29%3D0.3%2C%5C%20P%28E%7CC%29%3D0.4%2C%5C%20P%28E%7CD%29%3D0.6)
First we calculate the probability that print quality is poor,
![P(E)=P(A)P(E|A)+P(B)P(E|B)+P(C)P(E|C)+P(D)P(E|D)](https://tex.z-dn.net/?f=P%28E%29%3DP%28A%29P%28E%7CA%29%2BP%28B%29P%28E%7CB%29%2BP%28C%29P%28E%7CC%29%2BP%28D%29P%28E%7CD%29)
![P(E)=(0)(0.8)+(0.3)(0.02)+(0.4)(0.08)+(0.6)(0.1)](https://tex.z-dn.net/?f=P%28E%29%3D%280%29%280.8%29%2B%280.3%29%280.02%29%2B%280.4%29%280.08%29%2B%280.6%29%280.1%29)
![P(E)=0+0.006+0.032+0.06](https://tex.z-dn.net/?f=P%28E%29%3D0%2B0.006%2B0.032%2B0.06)
![P(E)=0.098](https://tex.z-dn.net/?f=P%28E%29%3D0.098)
a. Determine the probability of high ink viscosity given poor print quality.
![P(C|E)=\frac{P(E|C)P(C)}{P(E)}](https://tex.z-dn.net/?f=P%28C%7CE%29%3D%5Cfrac%7BP%28E%7CC%29P%28C%29%7D%7BP%28E%29%7D)
![P(C|E)=\frac{0.4\times 0.08}{0.098}](https://tex.z-dn.net/?f=P%28C%7CE%29%3D%5Cfrac%7B0.4%5Ctimes%200.08%7D%7B0.098%7D)
![P(C|E)=\frac{0.032}{0.098}](https://tex.z-dn.net/?f=P%28C%7CE%29%3D%5Cfrac%7B0.032%7D%7B0.098%7D)
![P(C|E)=0.3265](https://tex.z-dn.net/?f=P%28C%7CE%29%3D0.3265)
b. Given poor print quality, what problem is most likely?
Probability of no printer problem given poor quality is
![P(A|E)=\frac{P(E|A)P(A)}{P(E)}](https://tex.z-dn.net/?f=P%28A%7CE%29%3D%5Cfrac%7BP%28E%7CA%29P%28A%29%7D%7BP%28E%29%7D)
![P(A|E)=\frac{0\times 0.8}{0.098}](https://tex.z-dn.net/?f=P%28A%7CE%29%3D%5Cfrac%7B0%5Ctimes%200.8%7D%7B0.098%7D)
![P(A|E)=\frac{0}{0.098}](https://tex.z-dn.net/?f=P%28A%7CE%29%3D%5Cfrac%7B0%7D%7B0.098%7D)
![P(A|E)=0](https://tex.z-dn.net/?f=P%28A%7CE%29%3D0)
Probability of misaligned paper given poor quality is
![P(B|E)=\frac{P(E|B)P(B)}{P(E)}](https://tex.z-dn.net/?f=P%28B%7CE%29%3D%5Cfrac%7BP%28E%7CB%29P%28B%29%7D%7BP%28E%29%7D)
![P(B|E)=\frac{0.3\times 0.02}{0.098}](https://tex.z-dn.net/?f=P%28B%7CE%29%3D%5Cfrac%7B0.3%5Ctimes%200.02%7D%7B0.098%7D)
![P(B|E)=\frac{0.006}{0.098}](https://tex.z-dn.net/?f=P%28B%7CE%29%3D%5Cfrac%7B0.006%7D%7B0.098%7D)
![P(B|E)=0.0612](https://tex.z-dn.net/?f=P%28B%7CE%29%3D0.0612)
Probability of printer-head debris given poor quality is
![P(D|E)=\frac{P(E|D)P(D)}{P(E)}](https://tex.z-dn.net/?f=P%28D%7CE%29%3D%5Cfrac%7BP%28E%7CD%29P%28D%29%7D%7BP%28E%29%7D)
![P(D|E)=\frac{0.6\times 0.1}{0.098}](https://tex.z-dn.net/?f=P%28D%7CE%29%3D%5Cfrac%7B0.6%5Ctimes%200.1%7D%7B0.098%7D)
![P(D|E)=\frac{0.06}{0.098}](https://tex.z-dn.net/?f=P%28D%7CE%29%3D%5Cfrac%7B0.06%7D%7B0.098%7D)
![P(D|E)=0.6122](https://tex.z-dn.net/?f=P%28D%7CE%29%3D0.6122)
From the above conditional probabilities,
The printer-head debris problem is most likely given that print quality is poor.