In a factory manufacturing certain spare parts 2% of the output is found defective . If spares are packed in 200 units , the probability that there are 4 defective spares in a pack is

Options

A. `64 e^(-4)`
B. `32e^(-4)`
C. `(64)/(3) e^(-4)`
D. `(32)/(3) e^(-4)`

Answer

To solve the problem, we will use the Poisson distribution, which is appropriate for modeling the number of events (in this case, defective parts) in a fixed interval (in this case, a pack of 200 units).

### Step-by-Step Solution:

1. **Identify the parameters**:
- Given that 2% of the output is defective, we can calculate the expected number of defective parts in a pack of 200 units.
- The percentage of defective parts is 2%, which can be expressed as a decimal: \( p = \frac{2}{100} = 0.02 \).
- The total number of parts in a pack is \( n = 200 \).

2. **Calculate the expected number of defective parts (λ)**:
- The expected number of defective parts, denoted as \( \lambda \), is calculated as:
\[ \lambda = p \times n = 0.02 \times 200 = 4 \]

3. **Set up the Poisson probability formula**:
- The probability of observing \( k \) defects in a Poisson distribution is given by:
\[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]
- In this case, we want to find the probability of having \( k = 4 \) defective parts.

4. **Substitute the values into the formula**:
- We substitute \( \lambda = 4 \) and \( k = 4 \) into the formula:
\[ P(X = 4) = \frac{e^{-4} \cdot 4^4}{4!} \]

5. **Calculate \( 4^4 \) and \( 4! \)**:
- Calculate \( 4^4 = 256 \).
- Calculate \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

6. **Plug in the values**:
- Now we can substitute these values back into the probability formula:
\[ P(X = 4) = \frac{e^{-4} \cdot 256}{24} \]

7. **Simplify the expression**:
- Simplifying \( \frac{256}{24} \):
\[ \frac{256}{24} = \frac{256 \div 8}{24 \div 8} = \frac{32}{3} \]
- Thus, we have:
\[ P(X = 4) = \frac{32}{3} e^{-4} \]

8. **Final result**:
- The probability that there are 4 defective spares in a pack of 200 units is:
\[ P(X = 4) = \frac{32}{3} e^{-4} \]

### Conclusion:
The correct answer is \( \frac{32}{3} e^{-4} \).