In order to solve this, we'll use the properties of the Poisson distribution. For a Poisson random variable $X$ with mean $\mu$, the probability mass function (PMF) is given by:

$P(X = k) = \frac{\mu^k e^{-\mu}}{k!}$ where $\mu = 20$ and $k$ is the desired value of $X$.

We can also calculate cumulative probabilities using the cumulative distribution function (CDF) for the Poisson distribution.

a) $P(X \leq 11)$

This is the cumulative probability that $X$ takes on values from 0 to 11. We need to sum up the individual probabilities:

$P(X \leq 11) = \sum_{k=0}^{11} \frac{20^k e^{-20}}{k!}$

b) $P(X = 14)$

This is the probability of $X$ taking on the value 14:

$P(X = 14) = \frac{20^{14} e^{-20}}{14!}$

c) $P(X > 18)$

This is the complement of the cumulative probability up to 18:

$P(X > 18) = 1 - P(X \leq 18)$ We need to sum the probabilities for $X \leq 18$ and subtract from 1.

d) $P(19 \leq X \leq 27)$

This is the sum of the probabilities between 19 and 27:

$P(19 \leq X \leq 27) = \sum_{k=19}^{27} \frac{20^k e^{-20}}{k!}$

I will now calculate each of these probabilities using Python.Here are the calculated probabilities:

• a) $P(X \leq 11) = 0.0214$
• b) $P(X = 14) = 0.0387$
• c) $P(X > 18) = 0.6186$
• d) $P(19 \leq X \leq 27) = 0.5661$

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How would the probabilities change if $\mu$ were 30 instead of 20?
2. What is the expected value and variance of $X$ in the Poisson distribution?
3. How can the Poisson distribution be approximated using a normal distribution?
4. What is $P(X = 0)$ for this Poisson distribution with $\mu = 20$?
5. How would you calculate the cumulative probability for $P(X \leq 15)$?

Tip: When $\mu$ is large, the Poisson distribution can be approximated using a normal distribution with mean $\mu$ and variance $\mu$.