At a local community college, 56% of students who enter the college as freshmen go on to graduate. Eight freshmen are randomly selected.What is the probability that at most seven will graduate from the local community college?

Note: Do not round intermediate calculations. Round your final answer to 4 decimal places.

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.

At a local community college, 56% of students who enter the college as freshmen go on to graduate. Eight freshmen are randomly selected. What is the probability that at most seven will graduate from the local community college?

Calculate the probability that at most 7 out of 8 randomly selected freshmen will graduate from a local community college where the graduation rate is 56%. This problem uses the binomial distribution and covers key probability concepts. Suitable for high school students in Grades 11-12.

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