# 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 a certain set up, the machine produces 2 acceptable items, find the probability that the machine is correctly set up.

Let us assume U1, U2 and A be the events as follows:

U1 = Machine is correctly set up

U2 = Machine is incorrectly set up

A = produce two acceptable items

From the problem

P(A|U1) = P(producing 2 acceptable items if machine is correctly set up)

P(A|U2) = P(producing 2 acceptable items if machine is not correctly set up)

Now we find

P(U1|A) = P(Machine is correctly set up for producing 2 acceptable items)

Using Baye’s theorem:

The required probability is .

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