This problem can be solved using the binomial distribution formula. The binomial distribution is used when we have a fixed number of independent trials, each with two possible outcomes (success or failure). In this case:

• A "success" is a student graduating.
• The probability of success (graduation) is $p = 0.56$.
• The probability of failure (not graduating) is $q = 1 - p = 0.44$.
• The number of trials is $n = 8$ (since eight freshmen are selected).
• We want the probability that at most 7 students graduate, which is $P(X \leq 7)$.

Step 1: Define the binomial probability mass function (PMF)

The binomial probability for exactly $k$ successes (graduates) is given by the formula:

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

Where:

• $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$.
• $p^k$ is the probability of exactly $k$ successes.
• $(1 - p)^{n-k}$ is the probability of $n - k$ failures.

Step 2: Calculate the probability of at most 7 graduates

We need to calculate $P(X \leq 7)$, which is the sum of probabilities for 0 through 7 successes:

$P(X \leq 7) = P(X = 0) + P(X = 1) + \dots + P(X = 7)$

I will compute this value for you.The probability that at most 7 students will graduate from the local community college is approximately 0.9903.

Would you like further details on any part of the calculation?

Here are 5 related questions:

1. What is the probability that exactly 7 students will graduate?
2. How does the probability change if we select 10 freshmen instead of 8?
3. What is the probability that fewer than 3 students will graduate?
4. How would the probability change if the graduation rate were 70% instead of 56%?
5. Can the normal approximation to the binomial be applied in this case?

Tip: The binomial distribution is appropriate when you're dealing with events that have two possible outcomes and a fixed number of trials.