A production process produces an item. In a recent study, the company found that 20% of all items produced were defective. Each item is inspected before being shipped. 10% of the items were misclassified( labeled as defective when they were good or labeled good when they were defective). Construct a tree diagram and find the probability that a shipped item will be classified as good
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To construct a tree diagram, we need to break down the process into individual steps:
Step 1: Production process
- 80% chance of producing a good item (non-defective)
- 20% chance of producing a defective item
Step 2: Inspection process
- 90% chance of correctly classifying a good item
- 10% chance of misclassifying a good item as defective
- 90% chance of correctly classifying a defective item as defective
- 10% chance of misclassifying a defective item as good
Here is a graphical representation of the tree diagram:
Production
/ \
80% Good 20% Defective
/ \
Inspect Inspect
| / \
90% Good 10% Misclassified 90% Defective
| / \
Shipped Shipped
(Good) (Defective)
To find the probability that a shipped item will be classified as good, we need to consider all the paths that lead to a good classification at the "Shipped" stage.
Path 1: Production -> Inspect -> Shipped (Good)
- Probability: 80% (good production) * 90% (correct classification) = 72%
Path 2: Production -> Inspect -> Misclassified -> Shipped (Good)
- Probability: 80% (good production) * 10% (misclassification) * 90% (correct classification) = 7.2%
Adding the probabilities of both paths gives the overall probability that a shipped item will be classified as good:
72% + 7.2% = 79.2%
Therefore, the probability that a shipped item will be classified as good is 79.2%.