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

To construct a tree diagram, we need to consider the different outcomes or events that can occur.

Step 1: Start by drawing a vertical line to represent the main branch or the production process.

Step 2: Label the first branch as "Defective" and the second branch as "Good" since these are the possible outcomes of the production process.

Step 3: Attach probabilities to each branch. Since we know that 20% of all items produced are defective, we assign a probability of 0.20 to the "Defective" branch. Consequently, the "Good" branch will have a probability of 1 - 0.20 = 0.80 since it accounts for the complementary outcome.

Step 4: For the next set of branches, we need to consider the inspection process. The percentage of misclassification is given as 10%. Therefore, we assign a probability of 0.10 to the branch that represents misclassification, and the remaining 0.90 to the branch that represents accurate classification.

Here is how the tree diagram looks like:
```
| (0.80) Good
Production Process -|
| (0.20) Defective

| (0.90) Accurate Classification
Inspection Process -|
| (0.10) Misclassification

```

To find the probability that a shipped item will be classified as "Good," we need to consider the two possible paths that lead to this outcome:

1. The item being produced as "Good" (0.80 probability) and accurately classified (0.90 probability).
2. The item being produced as "Defective" (0.20 probability), but misclassified as "Good" (0.10 probability).

We need to calculate the probability for each path and then sum them up.

Probability that a shipped item will be classified as "Good" = (0.80) * (0.90) + (0.20) * (0.10) = 0.72 + 0.02 = 0.74

Therefore, the probability that a shipped item will be classified as "Good" is 0.74 or 74%.