A factory produces lazor blades and 1 in 500 blades is estimated to be defective. The blades are supplied in packets of 10. In a consignment of 10,000 packets, using poisson distribution, find approximately the number of packets which contain no defective blades.

Answer

To solve the problem step by step, we will use the Poisson distribution to find the number of packets that contain no defective blades.

### Step 1: Understand the problem
We know that:
- 1 in 500 blades is defective.
- Each packet contains 10 blades.
- We have a total of 10,000 packets.

### Step 2: Calculate the probability of a blade being defective
The probability \( P \) of a blade being defective is given by:
\[ P(\text{defective}) = \frac{1}{500} \]

### Step 3: Calculate the mean number of defective blades in a packet
Since each packet contains 10 blades, the mean number of defective blades \( \lambda \) in one packet can be calculated as:
\[ \lambda = n \times P = 10 \times \frac{1}{500} = \frac{10}{500} = \frac{1}{50} = 0.02 \]

### Step 4: Use the Poisson distribution formula
The Poisson distribution formula is given by:
\[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]
where \( k \) is the number of occurrences (defective blades in this case), and \( \lambda \) is the mean number of occurrences.

### Step 5: Find the probability of having no defective blades in a packet
We want to find the probability of having no defective blades, which means \( k = 0 \):
\[ P(X = 0) = \frac{e^{-0.02} \cdot 0.02^0}{0!} \]
Since \( 0! = 1 \) and \( 0.02^0 = 1 \), this simplifies to:
\[ P(X = 0) = e^{-0.02} \]

### Step 6: Calculate \( e^{-0.02} \)
Using the value of \( e^{-0.02} \):
\[ e^{-0.02} \approx 0.9802 \]

### Step 7: Calculate the expected number of packets with no defective blades
Now, we need to find the expected number of packets with no defective blades in 10,000 packets:
\[ \text{Expected number of packets} = 10,000 \times P(X = 0) = 10,000 \times 0.9802 \approx 9802 \]

### Final Answer
Thus, the approximate number of packets which contain no defective blades is **9802**.

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