If a machine is correctly set up, it produces 80% acceptable items. If it is incorrectly set up, it produces only 30% acceptable items. From the past experience it was known that 90% of the setups are correctly done. If after a certain setup, the machine produces 2 acceptable items then the probability that the machine was correctly set up, is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{64}{65} \)
Two acceptable items in a row have probability \( 0.8^2=0.64 \) for a correct setup and \( 0.3^2=0.09 \) for an incorrect one. By Bayes’ theorem,
\( \displaystyle P(\text{correct}\mid 2\ \text{acceptable})=\frac{0.9\times 0.64}{0.9\times 0.64+0.1\times 0.09} \)
\( \displaystyle =\frac{0.576}{0.585} \)
\( \displaystyle =\frac{64}{65} \)
NCERT (2026-27 edition):
Bayes Theorem
