Respuesta :
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.
[tex]P(B|A) = \frac{P(B)*P(A|B)}{P(A)}[/tex]
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 [tex]P(A) = 0.9402[/tex]
97% of the time the machine is set up correctly.
This means that [tex]P(B) = 0.97[/tex]
Furthermore, it is known that if the machine is set up correctly, it produces 96% acceptable (non-defective) items.
This means that [tex]P(A|B) = 0.96[/tex]
Probability:
[tex]P(B|A) = \frac{0.97*0.96}{0.9402} = 0.9904[/tex]
99.04% probability that the machine is set up correctly
