To solve Exercise 1 in the uploaded document, we need to calculate the probability that two defective items are produced in a run of 40 components using both the binomial distribution and its Poisson approximation.

Solution Using Binomial Distribution

Let $X$ be the number of defective components produced in a run of 40 components.

• Given: $X \sim B(40, 0.01)$
• We need to find $P(X = 2)$.

The probability mass function (PMF) of a binomial distribution is: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$ where:

• $n = 40$
• $k = 2$
• $p = 0.01$

Thus, $P(X = 2) = \binom{40}{2} (0.01)^2 (0.99)^{38}$

Calculating each part:

1. $\binom{40}{2} = \frac{40 \times 39}{2} = 780$
2. $(0.01)^2 = 0.0001$
3. $(0.99)^{38}$ (approximate or calculate precisely).

Plug these values in to get the final probability.

Solution Using Poisson Approximation

For large $n$ and small $p$, a binomial distribution $B(n, p)$ can be approximated by a Poisson distribution with parameter $\lambda = np$.

Here, $\lambda = 40 \times 0.01 = 0.4$.

For a Poisson random variable with mean $\lambda$, the probability of observing $k$ events is: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$ Thus, $P(X = 2) = \frac{(0.4)^2 e^{-0.4}}{2!}$

Calculating each part:

1. $(0.4)^2 = 0.16$
2. $e^{-0.4}$ (approximate or calculate precisely).
3. $2! = 2$

Combine these values for the final probability.

Would you like detailed calculations or explanations for each step?

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Additional Practice Questions:

1. How would the probability change if the machine’s defect rate increased to 2%?
2. How does the Poisson approximation perform as the defect rate increases?
3. Can you derive the formula for the Poisson distribution from the binomial distribution?
4. What assumptions are made when using the Poisson approximation?
5. How would you interpret a cumulative distribution for this problem?

Tip:

The Poisson approximation is especially useful when dealing with a large number of trials and a small probability of success.