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.

Answer 1

Let A be the event that the machine produces two acceptable items.
Also, let E₁ represent the event that the machine is correctly set up and E₂ represent the event that the machine is incorrectly set up

\[\therefore P\left( E_1 \right) = 0 . 8 \]

\[ P\left( E_2 \right) = 0 . 2\]

\[\text{ Now } , \]

\[P\left( A/ E_1 \right) = 0 . 9 \times 0 . 9 = 0 . 81\]

\[P\left( A/ E_2 \right) = 0 . 40 \times 0 . 40 = 0 . 16\]

\[\text{ Using Bayes' theorem, we get } \]

\[\text{ Required probability } = P\left( E_1 /A \right) = \frac{P\left( E_1 \right)P\left( A/ E_1 \right)}{P\left( E_1 \right)P\left( A/ E_1 \right) + P\left( E_2 \right)P\left( A/ E_2 \right)}\]

\[ = \frac{0 . 8 \times 0 . 81}{0 . 8 \times 0 . 81 + 0 . 2 \times 0 . 16}\]

\[ = \frac{81}{85}\]