A farmer has a tubewell which works by connecting two switches \( S_{1} \) and \( S_{2} \). The probability that \( S_{1} \) works is 9/10 and the probability that \( S_{2} \) works is 8/10. The farmer connects them in series in 30% cases and in parallel in 70% cases. If it is known that the tubewell was working, what is the probability that it was a series connection.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 108/451
This is a Bayes' theorem problem. Let E be the event that the tubewell works, \( E_{1} \) a series connection and \( E_{2} \) a parallel connection.
\( \displaystyle P(E_{1})=0.3 \)
\( \displaystyle P(E_{2})=0.7 \)
In series both switches must work. In parallel the tubewell fails only when both fail.
\( \displaystyle P(E\mid E_{1})=0.9\times0.8 \)
\( \displaystyle =0.72 \)
\( \displaystyle P(E\mid E_{2})=1-0.1\times0.2 \)
\( \displaystyle =0.98 \)
By Bayes' theorem:
\( \displaystyle P(E_{1}\mid E)=\frac{0.3\times0.72}{0.3\times0.72+0.7\times0.98} \)
\( \displaystyle =\frac{0.216}{0.902} \)
\( \displaystyle =\frac{108}{451} \)