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 setups are correctly done. If after certain setup the machine produces 2 acceptable items find the probability that the machine is correctly setup.

Answer 1

We will refer to "the probability of event X" by $P(X).$

As the question,
P(Correctly Setup)=$0.8$
P(Wrongly Setup)=$0.2$

The machine produces 2 acceptable items.

So, P(machine producing 2 acceptable items given that it is correctly setup) 
$=0.8 * (0.9)^2 = 0.8 * 0.81 = 0.648$

And, P(​machine producing 2 acceptable items given that it is wrongly setup),
$0.2 * (0.4)^2 = 0.2 * 0.16 = 0.032$

So, now P(machine correctly setup given that it produces 2 acceptable items) =

P(machine correctly setup given that it is correctly setup)/P(machine produces 2 acceptable items) 

=
$\frac{0.648}{0.648+0.032} = 0.9529$

Thus, there is a 95.29% probability that the machine was setup correctly given that it has produced 2 acceptable items