Question

A machine is used in a production process. From past data, it is known that 97% of the time the machine is set up correctly. Furthermore, it is known that if the machine is set up correctly, it produces 96% acceptable (non-defective) items. However, when it is set up incorrectly, it produces only 30% acceptable items.a. An item from the production line is selected. What is the probability that the selected item is non-defective? b. Given that the selected item is non-defective, what is the probability that the machine is set up correctly?

Answer 1

Answer:

a) 94.02% probability that the selected item is non-defective

b) 99.04% probability that the machine is set up correctly

Step-by-step explanation:

The Bayes Theorem is important to solve this question.

Bayes Theorem:

Two events, A and B.

$P(B|A) = \frac{P(B)*P(A|B)}{P(A)}$

In which P(B|A) is the probability of B happening when A has happened and P(A|B) is the probability of A happening when B has happened.

a. An item from the production line is selected. What is the probability that the selected item is non-defective?

The machine is set up correctly 97% of the time. When it is set up correctly, 96% of the items are non-defective.

The other 100-97 = 3% of the time, the machine is set up incorrectly. Then, 30% of the items are non-defective.

So

P = 0.97*0.96 + 0.03*0.3 = 0.9402

94.02% probability that the selected item is non-defective.

b. Given that the selected item is non-defective, what is the probability that the machine is set up correctly?

Then, for the Bayes Theorem:

Event A: The item is non-defective.

Event B: Machine set up correctly.

94.02% probability that the selected item is non-defective.

This means that $P(A) = 0.9402$

97% of the time the machine is set up correctly.

This means that $P(B) = 0.97$

Furthermore, it is known that if the machine is set up correctly, it produces 96% acceptable (non-defective) items.

This means that $P(A|B) = 0.96$

Probability:

$P(B|A) = \frac{0.97*0.96}{0.9402} = 0.9904$

99.04% probability that the machine is set up correctly