Question

A manufacturing machine has a 3% defect rate.

If 8 items are chosen at random, what is the probability that at least one will have a defect?

Answer 1

To find the probability that at least one item will have a defect when 8 items are selected at random from a manufacturing process with a 3% defect rate, we can use the complement rule.

1. **Define the scenario**:
- Probability of an item being defective (p) = 0.03
- Probability of an item being non-defective (q) = 1 - p = 1 - 0.03 = 0.97

2. **Calculate the probability that all 8 items are non-defective**:
The probability that all 8 chosen items are non-defective is given by:
\[
P(\text{all non-defective}) = q^n = 0.97^8
\]
where \( n = 8 \).

Now, we will calculate \( 0.97^8 \):
\[
0.97^8 \approx 0.783
\]

3. **Calculate the probability of having at least one defective item**:
The probability that at least one item is defective is the complement of all items being non-defective:
\[
P(\text{at least one defective}) = 1 - P(\text{all non-defective})
\]
\[
P(\text{at least one defective}) = 1 - 0.97^8 \approx 1 - 0.783 \approx 0.217
\]

Therefore, the probability that at least one of the 8 randomly chosen items will have a defect is approximately **0.217**, or **21.7%**.