a machine used in a production process is set up correctly 97% of the time. Given that the machine is set up correctly, only 5% of the items it produces are defective, while 95% are not defective. On the other hand, if the machine is set up incorrectly, 60% of the items it produces are defective, while only 40% are not defective. What is the probability that a randomly selected item from the production process will not be defective?
(0.97*0.95)+(0.03*0.40)
0.9335
To find the probability that a randomly selected item from the production process will not be defective, we need to consider two scenarios:
1. The machine is set up correctly.
2. The machine is set up incorrectly.
Let's calculate the probabilities for each scenario and then combine them using the law of total probability.
1. The machine is set up correctly:
Given that the machine is set up correctly, the probability of an item being defective is 5% (0.05), and the probability of an item not being defective is 95% (0.95).
2. The machine is set up incorrectly:
Given that the machine is set up incorrectly, the probability of an item being defective is 60% (0.60), and the probability of an item not being defective is 40% (0.40).
Now, let's calculate the overall probability of an item not being defective using the law of total probability:
P(not defective) = P(set up correctly) * P(not defective | set up correctly) + P(set up incorrectly) * P(not defective | set up incorrectly)
P(not defective) = 0.97 * 0.95 + (1 - 0.97) * 0.40
P(not defective) = 0.9225 + 0.03
P(not defective) = 0.9525
Therefore, the probability that a randomly selected item from the production process will not be defective is 0.9525, or 95.25%.