On an average 1 in 100 razor blades manufactured by a firm are defective.If blades are supplied in packets of 5 each, find the probability that a packet has atleast one defective blade. If 1000 packets are tested, in how many packets would you expect defective blades.

Answer

To solve the problem step by step, we will first find the probability that a packet of 5 razor blades contains at least one defective blade. Then, we will calculate how many packets out of 1000 are expected to contain at least one defective blade.

### Step 1: Determine the probability of a blade being defective
Given that 1 in 100 razor blades are defective, we can express this as:
- Probability of a blade being defective, \( p = \frac{1}{100} = 0.01 \)
- Probability of a blade being non-defective, \( q = 1 - p = 1 - 0.01 = 0.99 \)

### Step 2: Calculate the probability of a packet having no defective blades
A packet contains 5 blades. The probability that all 5 blades are non-defective is given by:
\[ P(\text{no defective blades}) = q^5 = (0.99)^5 \]

Calculating \( (0.99)^5 \):
\[ (0.99)^5 \approx 0.95099 \]

### Step 3: Calculate the probability of a packet having at least one defective blade
The probability that at least one blade in the packet is defective is:
\[ P(\text{at least one defective blade}) = 1 - P(\text{no defective blades}) = 1 - (0.99)^5 \]
\[ P(\text{at least one defective blade}) \approx 1 - 0.95099 \approx 0.04901 \]

### Step 4: Calculate the expected number of packets with at least one defective blade in 1000 packets
To find the expected number of packets with at least one defective blade when testing 1000 packets, we multiply the probability of having at least one defective blade by the total number of packets:
\[ \text{Expected number of defective packets} = 1000 \times P(\text{at least one defective blade}) \approx 1000 \times 0.04901 \]
\[ \text{Expected number of defective packets} \approx 49.01 \]

Since we cannot have a fraction of a packet, we can round this to approximately 49 packets.

### Final Answer
Thus, the probability that a packet has at least one defective blade is approximately \( 0.04901 \), and the expected number of packets with at least one defective blade out of 1000 packets is approximately 49.

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