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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability1 Mark
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.
B1 : Machine is correctly setup.
B2 : Machie is incorrectly setup.
Now P(B1) = 0.8, P(B2) = 0.2
P(A|B1) = 0.9 $\times$ 0.9 and P(A|B2) = 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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