Answered

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Suppose that only two factories make Playstation machines. Factory 1 produces 70% of the machines and Factory 2 produces the remaining 30%. Of the machines produced in Factory 1, 2% are defective. Of the machines produced in Factory 2, 5% are defective. What proportion of Playstation machines produced by these two factories are defective? Suppose that you purchase a playstation machine and it is defective. What is the probability that it was produced by Factory 1?

Sagot :

Given:

Factory 1 produces 70%

Factor 2 produces 30%

Defective machines in factory 1 = 2%

Defective machines in factory 2 = 5%

Find-:

What is the probability that it was produced by Factory 1?

Explanation-:

Probability of machines produced by factory1

[tex]\begin{gathered} P(F_1)=70\% \\ \\ P(F_1)=\frac{70}{100} \\ \\ P(F_1)=\frac{7}{10} \\ \end{gathered}[/tex]

Probability of machines produced by factory 2

[tex]\begin{gathered} P(F_2)=30\% \\ \\ P(F_2)=\frac{30}{100} \\ \\ P(F_2)=\frac{3}{10} \end{gathered}[/tex]

Probability of factory 1 produced defective item,

[tex]\begin{gathered} P(\frac{x}{F_1})=2\% \\ \\ P(\frac{x}{F_1})=\frac{2}{100} \\ \\ P(\frac{x}{F_1})=\frac{1}{50} \end{gathered}[/tex]

Probability of factory 2 produced defective item,

[tex]\begin{gathered} P(\frac{x}{F_2})=5\% \\ \\ P(\frac{x}{F_2})=\frac{5}{100} \\ \\ P(\frac{x}{F_2})=\frac{1}{20} \end{gathered}[/tex]

So, the probability that randomly selected items was form factor 1.

[tex]P(\frac{F_1}{x})\text{ is}[/tex]

Now, apply Bayes theorem is:

[tex]P(\frac{F_1}{x})=\frac{P(F_1)P(\frac{x}{F_1})}{P(F_1)P(\frac{x}{F_1})+P(F_2)P(\frac{x}{F_2})}[/tex]

So, the value is:

[tex]\begin{gathered} =\frac{\frac{7}{10}\times\frac{1}{50}}{\frac{7}{10}\times\frac{1}{50}+\frac{3}{10}\times\frac{1}{20}} \\ \\ =\frac{\frac{7}{500}}{\frac{7}{500}+\frac{3}{200}} \\ \\ =\frac{\frac{7}{5}}{\frac{7}{5}+\frac{3}{2}} \\ \\ =\frac{\frac{7}{5}}{\frac{14}{10}+\frac{15}{10}} \\ \\ =\frac{\frac{7}{5}}{\frac{14+15}{10}} \\ \\ =\frac{7}{5}\times\frac{10}{29} \\ \\ =\frac{14}{29} \end{gathered}[/tex]

So, the probability is 14/29.



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