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Part 1
​Hewlett-Packard (HP) receives large shipments of microprocessors from Intel Corp. It must try to ensure the proportion of microprocessors that are defective is small. Suppose HP decides to test eighteight microprocessors out of a shipment of thousands of these microprocessors. Suppose that if at least one of the microprocessors is​ defective, the shipment is returned. Complete parts a through c.
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Part 1
a. If Intel​ Corp.'s shipment contains 1212​% defective​ microprocessors, calculate the probability the entire shipment will be returned.
The probability is enter your response here.

We are given that Hewlett-Packard (HP) tests 8 microprocessors out of a shipment. The shipment contains 12% defective microprocessors, meaning the probability of a single microprocessor being defective is 0.12. HP returns the entire shipment if at least one of the microprocessors tested is defective.

We need to calculate the probability that the shipment will be returned, which occurs if at least one of the microprocessors is defective. First, we can calculate the probability that none of the 8 microprocessors is defective, and then subtract that from 1 to find the probability that at least one is defective.

### Steps:

1. **Probability that a single microprocessor is not defective**:  
   \( P(\text{not defective}) = 1 - 0.12 = 0.88 \)

2. **Probability that all 8 microprocessors are not defective**:  
   Since each test is independent, the probability that all 8 microprocessors are not defective is:  
   \[
   P(\text{all 8 not defective}) = 0.88^8
   \]

3. **Probability that at least one is defective**:  
   The probability that at least one of the 8 microprocessors is defective is:  
   \[
   P(\text{at least one defective}) = 1 - P(\text{all 8 not defective}) = 1 - 0.88^8
   \]

Let's calculate this probability.The probability that the entire shipment will be returned, meaning at least one of the 8 microprocessors is defective, is approximately **0.6404** or **64.04%**.

Would you like more details, or do you have any questions?

Here are 5 related questions to further explore:

1. What would happen to the return probability if only 5% of microprocessors were defective?
2. How does the probability change if HP tests 10 microprocessors instead of 8?
3. What is the probability of exactly two defective microprocessors in a test of 8?
4. How would the return probability change if HP decided to return the shipment only if two or more microprocessors were defective?
5. How can this approach be applied to other types of defect analysis?

**Tip**: The complement rule is a powerful tool in probability—use it whenever calculating "at least one" events!

Hewlett-Packard (HP) receives large shipments of microprocessors from Intel Corp. It must try to ensure the proportion of microprocessors that are defective is small. Suppose HP decides to test eight microprocessors out of a shipment of thousands of these microprocessors. Suppose that if at least one of the microprocessors is defective, the shipment is returned. Part a: If Intel Corp.'s shipment contains 12% defective microprocessors, calculate the probability the entire shipment will be returned.

Learn how to calculate the probability that a shipment of microprocessors will be returned if it contains 12% defective units. This problem covers key concepts such as probability, complement rule, and independent events. Detailed step-by-step solutions included for students at the college level or advanced high school.

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