Suppose a shipment of 180 electronic components contains 4 defective components. To determine whether the shipment should be​ accepted, a​ quality-control engineer randomly selects 4 of the components and tests them. If 1 or more of the components is​ defective, the shipment is rejected. What is the probability that the shipment is​ rejected?
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Part 1
The probability that the shipment is rejected is 
  
enter your response here.
​(Round to four decimal places as​ needed.)

To determine the probability that the shipment is rejected, we can calculate the probability that none of the selected components are defective (since the shipment is rejected if at least one defective component is found).

Let's define the problem in steps:

1. **Total Components**: There are 180 components in total.
2. **Defective Components**: Out of these, 4 are defective.
3. **Non-Defective Components**: This means there are \(180 - 4 = 176\) non-defective components.
4. **Selection**: We are selecting 4 components randomly without replacement.

To find the probability that the shipment is **not rejected**, we want the probability that all 4 selected components are non-defective. Then, we’ll subtract this result from 1 to find the probability of rejecting the shipment.

### Step-by-Step Calculation:

#### Step 1: Probability of Selecting 4 Non-Defective Components
The probability of selecting all non-defective components in a sample of 4 is calculated as:

\[
\frac{\binom{176}{4}}{\binom{180}{4}}
\]

where:
- \(\binom{176}{4}\) is the number of ways to choose 4 non-defective components from 176.
- \(\binom{180}{4}\) is the total number of ways to choose 4 components from 180.

#### Step 2: Calculating Each Part
1. **Calculate** \(\binom{176}{4}\):
   \[
   \binom{176}{4} = \frac{176!}{4!(176 - 4)!} = 54517776
   \]

2. **Calculate** \(\binom{180}{4}\):
   \[
   \binom{180}{4} = \frac{180!}{4!(180 - 4)!} = 68715552
   \]

#### Step 3: Find the Probability of Selecting All Non-Defective Components
Now we compute the probability that all selected components are non-defective:

\[
\frac{\binom{176}{4}}{\binom{180}{4}} = \frac{54517776}{68715552} \approx 0.7931
\]

#### Step 4: Find the Probability of Rejecting the Shipment
The probability that the shipment is rejected is the complement of the probability of selecting all non-defective components:

\[
1 - 0.7931 = 0.2069
\]

### Final Answer
The probability that the shipment is rejected is approximately:

\[
\boxed{0.2069}
\]

Let me know if you need any clarifications or further breakdown of the steps!

---

1. What if the shipment had a different number of defective components?
2. How does the sample size affect the probability of rejection?
3. What if the quality-control engineer tested more than 4 components?
4. How would the probability change if there were 200 components in total?
5. What is the probability that exactly one defective component is found in the sample?

**Tip**: Understanding combinations is essential for probability calculations involving "choose" scenarios.

Suppose a shipment of 180 electronic components contains 4 defective components. To determine whether the shipment should be accepted, a quality-control engineer randomly selects 4 of the components and tests them. If 1 or more of the components is defective, the shipment is rejected. What is the probability that the shipment is rejected?

Learn to calculate the probability of rejecting a shipment based on quality-control testing. This example uses combinatorics and the complement rule to determine the probability of finding defective components in a random sample. Suitable for high school students and those studying advanced probability.

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