An item is produced in three factories A*A*, B*B* and C*C*. Factory A*A* produces 2 times the number of items produced by factory B*B*, and the factories B*B* and C*C* produces the same number of items. It is known that 8%, 7%, 3% of the items produced by factories A*A*, B*B* and C*C* respectively are defective. All items produced in the three factories are stocked, and an item is selected at random. What is the probability that the selected item is defective? (Enter the answer correct to two decimal places)


To solve this problem, we need to find the probability that a randomly selected item from the combined production of the three factories is defective. We'll use the law of total probability.

### Step 1: Define Variables

Let's denote:
- \( N_A \) as the number of items produced by factory \( A \),
- \( N_B \) as the number of items produced by factory \( B \),
- \( N_C \) as the number of items produced by factory \( C \).

We know that:
\[ N_A = 2N_B \]
\[ N_B = N_C \]

### Step 2: Define Defect Probabilities

The probabilities that an item produced by each factory is defective are:
- \( P(D \mid A) = 0.08 \) (8% of items from \( A \) are defective),
- \( P(D \mid B) = 0.07 \) (7% of items from \( B \) are defective),
- \( P(D \mid C) = 0.03 \) (3% of items from \( C \) are defective).

### Step 3: Calculate the Total Production

Let \( N \) be the total production from all factories:
\[ N = N_A + N_B + N_C \]

Substituting the relationships between the productions:
\[ N = 2N_B + N_B + N_B = 4N_B \]

### Step 4: Calculate the Proportions of Production

The proportions of the total production from each factory are:
- From \( A \): \( \frac{N_A}{N} = \frac{2N_B}{4N_B} = \frac{1}{2} \),
- From \( B \): \( \frac{N_B}{N} = \frac{N_B}{4N_B} = \frac{1}{4} \),
- From \( C \): \( \frac{N_C}{N} = \frac{N_B}{4N_B} = \frac{1}{4} \).

### Step 5: Use the Law of Total Probability

The total probability that a randomly selected item is defective is given by:
\[ P(D) = P(A)P(D \mid A) + P(B)P(D \mid B) + P(C)P(D \mid C) \]

Substitute the known values:
\[ P(D) = \left(\frac{1}{2} \times 0.08\right) + \left(\frac{1}{4} \times 0.07\right) + \left(\frac{1}{4} \times 0.03\right) \]

Calculate each term:
\[ P(D) = 0.04 + 0.0175 + 0.0075 \]

Sum these probabilities:
\[ P(D) = 0.04 + 0.0175 + 0.0075 = 0.065 \]

Thus, the probability that a randomly selected item is defective is:
\[ \boxed{0.07} \]

Would you like more details or have any questions?

### Related Questions:

1. What is the probability that a non-defective item is selected?
2. How does the probability change if factory \( A \)'s defect rate drops to 5%?
3. If an item is found to be defective, what is the probability that it came from factory \( B \)?
4. How many defective items are produced by each factory if \( N_B = 1000 \)?
5. If factory \( C \) doubles its production, how does it affect the overall defect probability?
6. How would the probability change if the total production of all factories increased by 50%?
7. What is the expected number of defective items in a batch of 10000 items produced?
8. How do changes in individual defect rates impact the overall defect probability?

### Tip:

When solving probability problems involving multiple sources or conditions, clearly break down the contributions from each source and use the law of total probability for accurate results.

Calculate the probability that a randomly selected item from factories A, B, and C is defective using the law of total probability. Includes detailed steps on how to find the total production and proportions of defective items from each factory.

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