I have taken this problem from the paper of MASTER OF COMMERCE Examination of university in India with little modifications.
If a machine is correctly set up it will produce 90% acceptable items. If it is incorrectly set up it will produce 40% acceptable items. Past 5 years experience shows 80% of the setups are correctly done. If after a certain setup
1) machine produces 3 acceptable items and 1 non-acceptable( defective ) item as the first 4 pieces, find the probability that machine is correctly setup.
2)machine produces 3 acceptable items as the first 3 pieces,find the probability that the machine is correctly setup.
Answer: I have calculated the answer for 1)0.39724128 and for 2)0.99310345
I don't know what are the correct answers.So verification of these answers is needed.
Let us define the events:
The set up was correct.
The set up was wrong.
E=The item is acceptable.
D=The item is not acceptable (defective)
We know Probability that the set up was correct =0.8
Probability that the set up was wrong =0.2
Probability the item is acceptable given the information that the set up is correct =0.9
We are required to find the probability that the set up is correct given that the 3 items are acceptable, i-e
From the above calculations, i think you will know the calculations i have made for the first answer.