Given that x has a Poisson distribution with muequals0.1​, what is the probability that xequals4​?
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Part 1
​P(4​)almost equals

enter your response here ​(Round to four decimal places as​ needed.)

To solve this problem, we need to calculate the probability \( P(X = 4) \) for a Poisson distribution with a mean \( \mu = 0.1 \).

The formula for the Poisson probability mass function is:

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

Where:
- \( \mu = 0.1 \) (mean),
- \( k = 4 \) (we want the probability that \( X = 4 \)),
- \( e \) is the base of the natural logarithm (approximately 2.71828).

Let's compute it step by step:

\[
P(X = 4) = \frac{e^{-0.1} \cdot 0.1^4}{4!}
\]

1. \( e^{-0.1} \approx 0.9048 \)
2. \( 0.1^4 = 0.0001 \)
3. \( 4! = 24 \)

Now, substitute these values:

\[
P(X = 4) = \frac{0.9048 \times 0.0001}{24} \approx \frac{0.00009048}{24} = 0.00000377
\]

Thus, the probability \( P(X = 4) \approx 0.0000038 \).

### Final Answer:
\[
P(X = 4) \approx 0.0000038
\]

If you have any further questions, feel free to ask!

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Here are some related questions to consider:
1. What is the probability that \( X = 2 \) for the same Poisson distribution?
2. How does changing the mean \( \mu \) affect the shape of a Poisson distribution?
3. What is the cumulative probability \( P(X \leq 4) \) in this scenario?
4. How would you compute \( P(X = 0) \) for \( \mu = 0.1 \)?
5. Can you compare the Poisson distribution to the binomial distribution for small values of \( \mu \)?

**Tip**: The Poisson distribution is often used for modeling rare events, so when \( \mu \) is small, the probability of higher values of \( X \) becomes very small.

Given that x has a Poisson distribution with μ = 0.1, what is the probability that x = 4? (Round to four decimal places as needed.)

Find the probability of x = 4 in a Poisson distribution with a mean of 0.1. This problem involves calculating the probability using the Poisson probability mass function, including step-by-step breakdowns. Suitable for students in advanced high school or undergraduate probability courses.

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