A manufacturing machine has a 6% defect rate.

If 8 items are chosen at random, what is the probability that at least one will have a defect? Give your answer as a fraction or decimal out to at least 4 places.

To find the probability that at least one of the 8 items chosen will have a defect, we can approach this problem using the complement rule. The complement rule states that the probability of an event happening is \(1 - \) the probability of it not happening.

### Step-by-step Solution:

1. **Defining the Problem**: 
   The probability of an item being defective is 6%, or \(0.06\). Therefore, the probability that an item is **not defective** is \(1 - 0.06 = 0.94\).

2. **Finding the Complement**:
   The probability that **none** of the 8 items are defective (i.e., all 8 items are non-defective) is the product of the probability that each individual item is non-defective:
   \[
   P(\text{none defective}) = (0.94)^8
   \]

3. **Calculation**:
   Now, calculate \(0.94^8\):
   \[
   0.94^8 \approx 0.6243
   \]

4. **Finding the Desired Probability**:
   The probability that **at least one** of the 8 items is defective is the complement of the probability that none are defective:
   \[
   P(\text{at least one defective}) = 1 - P(\text{none defective}) = 1 - 0.6243 = 0.3757
   \]

Thus, the probability that at least one of the 8 items is defective is approximately \(0.3757\).

### Final Answer:
\[
P(\text{at least one defective}) \approx 0.3757
\]

Would you like further details or have any other questions?

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Here are 5 related questions you might find interesting:
1. What is the probability that exactly two of the items are defective?
2. How would the probability change if the defect rate increased to 10%?
3. If the sample size increased to 12 items, what would be the probability of at least one defect?
4. What is the probability that none of the 8 items are defective?
5. How do probabilities change when choosing items with replacement versus without?

**Tip**: Understanding complements is a powerful tool in probability, especially when calculating "at least one" events!

A manufacturing machine has a 6% defect rate. If 8 items are chosen at random, what is the probability that at least one will have a defect? Give your answer as a fraction or decimal out to at least 4 places.

Calculate the probability that at least one item out of 8 randomly chosen items is defective, given a 6% defect rate. This step-by-step solution explains the complement rule, key probability concepts, and provides a detailed calculation. Suitable for students in Grades 10-12.

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