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?

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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!