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

Answer 1

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(U₁) = 0.8

⇒ P(U₂) = 0.2

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

⇒ P(A|U₁) = 0.9 x 0.9

⇒ P(A|U₁) = 0.81

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

⇒ P(A|U₂) = 0.4 x 0.4

⇒ P(A|U₂) = 0.16

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 \(\cfrac{81}{85}.\)