This problem deals with binomial probability because we're given a fixed number of trials (44 students), a binary outcome (whether a student needs to take another math class or not), and a constant probability (60% of students still need to take another math class).

Let's break down the steps to solve each part:

Given:

• Probability that a student needs to take another math class, $p = 0.60$
• Number of students, $n = 44$
• We are asked to compute probabilities for various outcomes.

We will use the binomial probability formula: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$ Where:

• $X$ is the number of students who need to take another math class.
• $k$ is the desired number of students.
• $\binom{n}{k}$ is the binomial coefficient.

For the cumulative probabilities, we'll sum probabilities for different values.

a. Probability that exactly 28 of them need to take another math class.

We calculate $P(X = 28)$.

b. Probability that at most 27 of them need to take another math class.

This is the cumulative probability: $P(X \leq 27) = P(X = 0) + P(X = 1) + \dots + P(X = 27)$

c. Probability that at least 27 of them need to take another math class.

This is the complementary probability: $P(X \geq 27) = 1 - P(X \leq 26)$

d. Probability that between 25 and 30 of them (inclusive) need to take another math class.

We sum the probabilities for $P(25 \leq X \leq 30)$: $P(25 \leq X \leq 30) = P(X = 25) + P(X = 26) + P(X = 27) + P(X = 28) + P(X = 29) + P(X = 30)$

I'll compute these probabilities now.Here are the calculated probabilities, rounded to four decimal places as required:

• a. Probability that exactly 28 of them need to take another math class: $0.1099$
• b. Probability that at most 27 of them need to take another math class: $0.6288$
• c. Probability that at least 27 of them need to take another math class: $0.4919$
• d. Probability that between 25 and 30 (inclusive) of them need to take another math class: $0.6206$

Would you like further explanation on any part of these calculations?

Here are 5 related questions you might find useful:

1. How does the binomial distribution relate to the normal distribution for large $n$?
2. What is the difference between cumulative and point probabilities in binomial distributions?
3. How would the probabilities change if 70% of students needed another math class instead of 60%?
4. How do you calculate the binomial coefficient $\binom{n}{k}$ manually?
5. What is the expected number of students who need to take another math class?

Tip: For large binomial problems, approximating the distribution with a normal distribution can simplify the calculations.