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.