
b. What is the probability of finding at most 5 defects per shipment?
c. Determine the average number of defects found in a shipment.

To solve these questions, I'll assume we are working with a **Poisson distribution**, which is commonly used for modeling the number of defects in a shipment when they occur randomly and independently. In a Poisson distribution, the probability of a given number of defects is determined by the formula:

\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
\]

Where:
- \(X\) is the random variable representing the number of defects,
- \(\lambda\) is the average number of defects per shipment,
- \(e\) is Euler's number (approximately 2.718),
- \(k\) is the number of defects.

Now let's address your specific questions:

### (b) **Probability of finding at most 5 defects per shipment**:
The probability of finding "at most 5 defects" means we need to calculate:

\[
P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
\]

To do this, we need the value of \(\lambda\) (the average number of defects per shipment). If you can provide that value, we can calculate the probabilities for each \(P(X = k)\) term and sum them up.

### (c) **Average number of defects found in a shipment**:
In a Poisson distribution, the average (or expected) number of defects is simply the value of \(\lambda\). 

If \(\lambda\) is given or can be calculated based on additional data, that would be the average number of defects per shipment.

Could you please provide the value of \(\lambda\), or more details, so I can complete these calculations?

---

**Questions for further exploration:**

1. How is the Poisson distribution applied in quality control processes?
2. What does \(\lambda\) represent in the context of defect counting?
3. How can you estimate \(\lambda\) if only limited data is available?
4. Can we approximate a Poisson distribution by a normal distribution if \(\lambda\) is large?
5. How does changing \(\lambda\) affect the spread and shape of the Poisson distribution?

**Tip:** In a Poisson distribution, the variance is also equal to \(\lambda\), meaning the spread of defects is closely tied to the average!

What is the probability of finding at most 5 defects per shipment? Determine the average number of defects found in a shipment.

Explore how to calculate the probability of finding at most 5 defects in a shipment using the Poisson distribution. Learn about the average number of defects and how to apply the Poisson formula for quality control analysis. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation