If a machine is correctly set up it produces 90 % acceptable item… - Vidyadip

Question

If a machine is correctly set up it produces $90\%$ acceptable items. If it is incorrectly set up it produces only $40\%$ acceptable item. Past experience shows that $80\%$ of the setups are correctly done. If after a certain set up, the machine produces $2$ acceptable items, find the probability that the machine is correctly set up.

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Answer

Let A be the event that the machine produces two acceptable items.
Also, let $E_1$ represent the event that the machine is correctly set up and $E_2$ represent the event that the machine is incorrectly set up
$\therefore\ \text{P}(\text{E}_1)=0.8$
$\text{P}(\text{E}_2)=0.2$
Now,
$\text{P}\Big(\frac{\text{A}}{\text{E}_1}\Big)=0.9\times0.9=0.81$
$\text{P}\Big(\frac{\text{A}}{\text{E}_2}\Big)=0.40\times0.40=0.16$
Using Bayes, theorem, we get
Required probability $=\text{P}\Big(\frac{\text{E}_1}{\text{A}}\Big)=\frac{\text{P}(\text{E}_1)\text{P}\Big(\frac{\text{A}}{\text{E}_1}\Big)}{\text{P}(\text{E}_1)\text{P}\Big(\frac{\text{A}}{\text{E}_1}\Big)+\text{P}(\text{E}_2)\text{P}\Big(\frac{\text{A}}{\text{E}_2}\Big)}$
$=\frac{0.8\times0.81}{0.8\times0.81+0.2\times0.61}$
$=\frac{81}{85}$

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