If a machine is correctly set up, it produces 90 % acceptable ite… - 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 items. Past experience shows that $80\%$ of the set ups are correctly done. If after a certain set up, the machine produces $2$ acceptable items, find the probability that the machine is correctly setup.

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Answer

Let's define events;
$A :$ Machine produces $2$ acceptable items.
$B_1:$ Machine is correctly setup.
$B_2:$ Machie is incorrectly setup.
Now $P(B_1) = 0.8, P(B_2) = 0.2$
$P(A|B_1) = 0.9 \times 0.9$ and $P(A|B_2) = 0.4 \times 0.4$
Therefore $\mathrm{P}\left(\mathrm{B}_{1} | \mathrm{A}\right)=\frac{\mathrm{P}\left(\mathrm{B}_{1}\right) \mathrm{P}\left(\mathrm{A} | \mathrm{B}_{1}\right)}{\mathrm{P}\left(\mathrm{B}_{1}\right) \mathrm{P}\left(\mathrm{A} | \mathrm{B}_{1}\right)+\mathrm{P}\left(\mathrm{B}_{2}\right) \mathrm{P}\left(\mathrm{A} | \mathrm{B}_{2}\right)}$
= $\frac{0.8 \times 0.9 \times 0.9}{0.8 \times 0.9 \times 0.9+0.2 \times 0.4 \times 0.4}$ = $\frac{648}{680}= \frac{81}{85}$

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