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 none of them graduates from the local community college?

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

This problem involves a binomial probability distribution, where we are given:

- The probability that a student graduates, \( p = 0.56 \),
- The number of trials (students), \( n = 8 \),
- The probability that a student does not graduate, \( q = 1 - p = 1 - 0.56 = 0.44 \).

We are asked to find the probability that **none** of the selected 8 students graduates. In other words, the number of successes \( x = 0 \).

The binomial probability formula is:

\[
P(X = x) = \binom{n}{x} p^x q^{n-x}
\]

For \( x = 0 \), this becomes:

\[
P(X = 0) = \binom{8}{0} (0.56)^0 (0.44)^8
\]

Since \( \binom{8}{0} = 1 \) and \( (0.56)^0 = 1 \), this simplifies to:

\[
P(X = 0) = (0.44)^8
\]

Let's calculate that.

\[
P(X = 0) = 0.44^8
\]The probability that none of the eight randomly selected freshmen graduates from the local community college is approximately \( 0.0014 \), or 0.14%.

Would you like more details or have any further questions?

Here are 5 related questions to explore:

1. What is the probability that exactly one student graduates?
2. How would the probability change if the graduation rate were 70% instead of 56%?
3. What is the probability that at least one student graduates?
4. How can we compute the expected number of graduates from these eight students?
5. What is the standard deviation of the number of graduates in this scenario?

**Tip:** For binomial problems, use the formula \( P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \) to find the probability of any specific number of successes!

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 none of them graduates from the local community college? Note: Do not round intermediate calculations. Round your final answer to 4 decimal places.

Solve a probability problem using the binomial distribution where 56% of students graduate. What is the probability that none of the 8 selected freshmen graduate? This example walks through the binomial probability formula and key concepts in probability theory.

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