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 solve this, we can use the concept of combinations and probabilities. The problem involves determining the probability that a shipment of 180 components is rejected if 1 or more of the selected 4 components is defective. Here's the step-by-step solution:

### Step 1: Total possible selections
The total number of ways to choose 4 components from the 180 is calculated using combinations:

\[
\binom{180}{4} = \frac{180 \times 179 \times 178 \times 177}{4 \times 3 \times 2 \times 1} = 6,916,635
\]

### Step 2: Favorable outcome (No defective component)
Next, we calculate the number of ways to select 4 non-defective components. Since there are 4 defective components, there are \(180 - 4 = 176\) non-defective components. The number of ways to select 4 components from the 176 non-defective ones is:

\[
\binom{176}{4} = \frac{176 \times 175 \times 174 \times 173}{4 \times 3 \times 2 \times 1} = 5,939,544
\]

### Step 3: Probability of no defective components
The probability of selecting 4 non-defective components (i.e., the shipment is **not rejected**) is the ratio of favorable outcomes to total outcomes:

\[
P(\text{no defective}) = \frac{\binom{176}{4}}{\binom{180}{4}} = \frac{5,939,544}{6,916,635} \approx 0.8590
\]

### Step 4: Probability of at least 1 defective component
The probability of selecting **at least 1 defective component** (which would cause the shipment to be rejected) is the complement of the probability of selecting no defective components:

\[
P(\text{rejected}) = 1 - P(\text{no defective}) = 1 - 0.8590 = 0.1410
\]

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

\[
\boxed{0.1410}
\]

Would you like more details on how combinations are calculated, or have any questions?

Here are 5 related questions to deepen your understanding:
1. What if the shipment had 5 defective components instead of 4? How would the probability change?
2. How would the probability change if the quality-control engineer tested 5 components instead of 4?
3. What is the probability that exactly 2 of the selected components are defective?
4. What is the probability that none of the components are defective if only 3 components are selected?
5. What is the probability that all selected components are defective?

**Tip:** For problems involving selection, combinations are often more appropriate than permutations when the order doesn't matter.

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? (Round to four decimal places as​ needed.)

This problem explores the probability of rejecting a shipment containing 4 defective components out of 180. Using combinations and the complement rule, it calculates the likelihood of detecting at least one defective component during a random selection of 4 components. This problem demonstrates practical applications of combinatorics and probability theory.

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