Suppose you have just received a shipment of 15 modems. Although you​ don't know​ this, 3 of the modems are defective. To determine whether you will accept the​ shipment, you randomly select 4 modems and test them. If all 4 modems​ work, you accept the shipment.​ Otherwise, the shipment is rejected. What is the probability of accepting the​ shipment?
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Part 1
The probability of accepting the shipment is 
  
enter your response here.
​(Round to four decimal places as​ needed.)

This problem is a classic example of using combinatorics to calculate probabilities. Here's how to approach it:

### Problem Recap:
- You have 15 modems in total.
- 3 modems are defective, and 12 modems are working.
- You randomly select 4 modems.
- The shipment is accepted only if all 4 modems work.

### Objective:
You need to calculate the probability of accepting the shipment, which happens if all 4 randomly selected modems are working.

### Approach:
1. **Total Number of Ways to Select 4 Modems**:  
   The total number of ways to choose 4 modems out of 15 is given by the combination formula \( C(n, k) \), which represents the number of ways to choose \( k \) objects from a set of \( n \) objects without regard to the order:
   \[
   C(15, 4) = \frac{15!}{4!(15-4)!} = \frac{15!}{4! \cdot 11!}
   \]

2. **Number of Ways to Select 4 Working Modems**:  
   Since there are 12 working modems, the number of ways to choose 4 working modems out of 12 is:
   \[
   C(12, 4) = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!}
   \]

3. **Probability of Accepting the Shipment**:  
   The probability of accepting the shipment (i.e., choosing 4 working modems) is the ratio of the number of favorable outcomes (selecting 4 working modems) to the total number of possible outcomes (selecting any 4 modems from the 15):
   \[
   P(\text{Accept}) = \frac{C(12, 4)}{C(15, 4)}
   \]

Let's compute this step-by-step.The probability of accepting the shipment, where all 4 selected modems work, is approximately \( 0.3626 \). 

This means there is a 36.26% chance that the shipment will be accepted.

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the probability change if there were only 10 modems instead of 15?
2. What is the probability of rejecting the shipment?
3. How would the probability change if you tested 5 modems instead of 4?
4. How can this probability be interpreted in a real-world scenario for quality control?
5. What is the expected number of defective modems in a random selection of 4?

**Tip:** Understanding combinatorics is crucial for solving many probability problems. Practicing with different values can strengthen your grasp of these concepts.

Explore the probability of accepting a shipment of 15 modems, where 3 are defective. Learn how to calculate the probability using combinatorics, focusing on selecting 4 modems and ensuring all are working. Approximate the chance of shipment acceptance with step-by-step guidance.

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