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.