Question

A manufacturing machine has a 10% defect rate.

If 9 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 has a defect when 9 items are chosen at random from a manufacturing machine with a 10% defect rate, we can use the complementary probability approach.

1. **Define the parameters**:
- Let \( p \) be the probability that an item is defective (which is 0.10 or 10%).
- Let \( q \) be the probability that an item is not defective (which is \( 1 - p = 0.90 \) or 90%).

2. **Calculate the complementary probability**:
- The complementary event to "at least one defect" is "no defects". To find the probability of no defects in 9 items, you would calculate \( q^n \) where \( n \) is the number of items (in this case, 9).

\[
P(\text{no defects}) = q^9 = 0.90^9
\]

3. **Calculate \( 0.90^9 \)**:
\[
0.90^9 \approx 0.3874
\]

4. **Calculate the probability of at least one defect**:
\[
P(\text{at least one defect}) = 1 - P(\text{no defects}) = 1 - 0.90^9
\]
\[
P(\text{at least one defect}) \approx 1 - 0.3874 \approx 0.6126
\]

Therefore, the probability that at least one of the 9 items is defective is approximately \( 0.6126 \), or 61.26%.