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 its 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? Construct a tree diagram
To solve this problem and construct a tree diagram, we will break it down into understandable steps:
Step 1: Construct the tree diagram:
- Start by drawing a line or branch to represent the machine being set up correctly.
- Label this branch as "Correct Setup" and write the probability of the machine being set up correctly (97%) next to the branch.
- Split this branch into two branches: one for defective items and one for non-defective items.
- Write the probability of a defective item (5%) next to the defective branch and the probability of a non-defective item (95%) next to the non-defective branch.
- From the "Correct Setup" branch, draw another branch to represent the machine being set up incorrectly.
- Label this branch as "Incorrect Setup" and write the probability of the machine being set up incorrectly (3%) next to the branch.
- Split this branch into two branches: one for defective items and one for non-defective items.
- Write the probability of a defective item (60%) next to the defective branch and the probability of a non-defective item (40%) next to the non-defective branch.
- You should now have a tree diagram with four branches in total.
Step 2: Calculate the probability of a randomly selected item being non-defective:
- Look at the branches representing non-defective items in the tree diagram.
- Add up the probabilities of these branches to find the overall probability of a non-defective item.
- In this case, you would add 0.97 x 0.95 (branch from "Correct Setup" -> non-defective item) and 0.03 x 0.40 (branch from "Incorrect Setup" -> non-defective item) to get the total probability.
Step 3: Calculate the probability:
- Multiply the probabilities calculated in step 2 to get the final probability.
- In this case, the final probability would be 0.97 x 0.95 + 0.03 x 0.40 = 0.9215.
Therefore, the probability that a randomly selected item from the production process will not be defective is approximately 0.9215, or 92.15%.