This problem involves a binomial distribution since each incoming call is either a fax (with probability $p = 0.25$) or not a fax (with probability $1 - p = 0.75$), and there is a fixed number of trials ($n = 25$).

Let $X$ be the number of fax messages among the 25 calls. Then, $X$ follows a binomial distribution: $X \sim \text{Binomial}(n = 25, p = 0.25)$

To solve each part, we use the binomial probability formula or cumulative probabilities, where:

$P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$

(a) Probability that at most 8 of the calls involve a fax message:

We want to calculate $P(X \leq 8)$, which is the cumulative probability from 0 up to 8.

$P(X \leq 8) = \sum_{k=0}^{8} P(X = k)$

Using binomial cumulative distribution functions (often available on calculators or software):

$P(X \leq 8) \approx 0.819$

(b) Probability that exactly 8 of the calls involve a fax message:

We want $P(X = 8)$, which is calculated using the binomial formula:

$P(X = 8) = \binom{25}{8} (0.25)^8 (0.75)^{17} \approx 0.159$

(c) Probability that at least 8 of the calls involve a fax message:

This is $P(X \geq 8)$, which can be calculated as:

$P(X \geq 8) = 1 - P(X \leq 7)$

Using cumulative probabilities:

$P(X \geq 8) \approx 1 - 0.726 = 0.274$

(d) Probability that more than 8 of the calls involve a fax message:

This is $P(X > 8)$, which can be calculated as:

$P(X > 8) = 1 - P(X \leq 8)$

Using cumulative probabilities:

$P(X > 8) \approx 1 - 0.819 = 0.181$

Summary of Answers:

1. $P(X \leq 8) \approx 0.819$
2. $P(X = 8) \approx 0.159$
3. $P(X \geq 8) \approx 0.274$
4. $P(X > 8) \approx 0.181$

Would you like further explanations on any of these steps or have additional questions?

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Here are some related questions you might find helpful:

1. What is the binomial distribution and when is it applicable?
2. How can we approximate binomial probabilities using the normal distribution?
3. What are the mean and variance of a binomial distribution?
4. How do cumulative distribution functions (CDFs) work for binomial probabilities?
5. What are other practical applications of the binomial distribution?

Tip: For large values of $n$ and probabilities not close to 0 or 1, binomial distributions can often be approximated by a normal distribution to simplify calculations.