Calculate each Poisson probability:

a. P(X = 9), λ = 14

To calculate the Poisson probability \( P(X = 9) \) with \( \lambda = 14 \), we use the Poisson probability formula:

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

Here:
- \( \lambda = 14 \)
- \( k = 9 \)
- \( e \approx 2.71828 \)

Now let's compute the probability.The Poisson probability \( P(X = 9) \) for \( \lambda = 14 \) is approximately **0.0473**.

Would you like more details on this calculation or have any further questions?

Here are five related questions to consider:
1. How does the Poisson distribution change when \( \lambda \) increases?
2. What is the cumulative probability \( P(X \leq 9) \) for the same \( \lambda = 14 \)?
3. How do you interpret the result of \( P(X = 9) \) in a real-world scenario?
4. How does the Poisson distribution compare to the binomial distribution for large \( \lambda \)?
5. How do you compute the mean and variance of a Poisson distribution?

**Tip:** The mean and variance of a Poisson distribution are both equal to \( \lambda \).

Learn how to calculate the Poisson probability P(X = 9) with λ = 14 using the Poisson probability formula. Understand the concepts of λ and k in the context of probability theory. Suitable for advanced high school students and anyone interested in probability calculations.

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