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.

$0.95$

The correct answer is Option (2) → $0.95$ ##

Let A be the event that the machine produces 2 acceptable items.

Also let $B_1$ represent the event of correct set up and $B_2$ represent the event of incorrect setup.

Now $P(B_1) = 0.8, P(B_2) = 0.2$

$P(A|B_1) = 0.9 \times 0.9 \text{ and } P(A|B_2) = 0.4 \times 0.4$

Therefore $P(B_1|A) = \frac{P(B_1) P(A|B_1)}{P(B_1) P(A|B_1) + P(B_2) P(A|B_2)}$

$= \frac{0.8 \times 0.9 \times 0.9}{0.8 \times 0.9 \times 0.9 + 0.2 \times 0.4 \times 0.4} = \frac{648}{680} = 0.95$