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
[tex]P(E|A)=0,\ P(E|B)=0.3,\ P(E|C)=0.4,\ P(E|D)=0.6[/tex]
First we calculate the probability that print quality is poor,
[tex]P(E)=P(A)P(E|A)+P(B)P(E|B)+P(C)P(E|C)+P(D)P(E|D)[/tex]
[tex]P(E)=(0)(0.8)+(0.3)(0.02)+(0.4)(0.08)+(0.6)(0.1)[/tex]
[tex]P(E)=0+0.006+0.032+0.06[/tex]
[tex]P(E)=0.098[/tex]
a. Determine the probability of high ink viscosity given poor print quality.
[tex]P(C|E)=\frac{P(E|C)P(C)}{P(E)}[/tex]
[tex]P(C|E)=\frac{0.4\times 0.08}{0.098}[/tex]
[tex]P(C|E)=\frac{0.032}{0.098}[/tex]
[tex]P(C|E)=0.3265[/tex]
b. Given poor print quality, what problem is most likely?
Probability of no printer problem given poor quality is
[tex]P(A|E)=\frac{P(E|A)P(A)}{P(E)}[/tex]
[tex]P(A|E)=\frac{0\times 0.8}{0.098}[/tex]
[tex]P(A|E)=\frac{0}{0.098}[/tex]
[tex]P(A|E)=0[/tex]
Probability of misaligned paper given poor quality is
[tex]P(B|E)=\frac{P(E|B)P(B)}{P(E)}[/tex]
[tex]P(B|E)=\frac{0.3\times 0.02}{0.098}[/tex]
[tex]P(B|E)=\frac{0.006}{0.098}[/tex]
[tex]P(B|E)=0.0612[/tex]
Probability of printer-head debris given poor quality is
[tex]P(D|E)=\frac{P(E|D)P(D)}{P(E)}[/tex]
[tex]P(D|E)=\frac{0.6\times 0.1}{0.098}[/tex]
[tex]P(D|E)=\frac{0.06}{0.098}[/tex]
[tex]P(D|E)=0.6122[/tex]
From the above conditional probabilities,
The printer-head debris problem is most likely given that print quality is poor.
Answer:
Answer of Part(a) is 16/49
and Answer of Part(b) is Printer-head debris
Step-by-step explanation:
Answer is in the following attachment
