A manufacturing machine has a 7% defect rate.

If 10 items are chosen at random, what is the probability that at least one will have a defect?

Round to three decimal places
  

To solve this problem, we’ll use the complement rule, which is easier here than directly calculating probabilities for one or more defective items.

### Step-by-Step Solution

1. **Define the Probability of No Defect**:  
   Since the defect rate is \(7\%\), the probability of selecting a non-defective item is:
   \[
   P(\text{no defect}) = 1 - 0.07 = 0.93
   \]

2. **Calculate the Probability of All 10 Items Being Non-Defective**:  
   If we choose 10 items at random, the probability that all of them are non-defective is:
   \[
   P(\text{all 10 non-defective}) = 0.93^{10}
   \]

3. **Calculate the Complement (At Least One Defective Item)**:  
   The probability of having at least one defective item is the complement of all 10 items being non-defective. So:
   \[
   P(\text{at least one defective}) = 1 - 0.93^{10}
   \]

4. **Compute the Result**:
   \[
   0.93^{10} \approx 0.484
   \]
   \[
   P(\text{at least one defective}) = 1 - 0.484 = 0.516
   \]

### Final Answer
The probability that at least one of the 10 items will have a defect is approximately:
\[
\boxed{0.516}
\]

Would you like further details, or do you have any questions about the steps?

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**Related Questions:**

1. What would the probability be if the defect rate was increased to 10%?
2. How does the probability change if only 5 items are selected instead of 10?
3. What is the probability of finding exactly one defective item out of 10?
4. How would you calculate the probability of exactly two defective items in a batch of 10?
5. How would this calculation differ if items were chosen one at a time with replacement?

**Tip:** Using the complement rule can often simplify calculations, especially when "at least one" is involved.

A manufacturing machine has a 7% defect rate. If 10 items are chosen at random, what is the probability that at least one will have a defect? Round to three decimal places.

Calculate the probability of finding at least one defective item out of 10 when a machine has a 7% defect rate. This solution uses the complement rule, providing a clear step-by-step approach. Perfect for students learning about probability and real-world applications.

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