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.

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Answers

2014-03-17T17:38:53+05:30
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
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