More than 96 percent of the very largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 13 such institutions. We are interested in the number that offer distance courses.  

Find the probability that at most ten offer such courses.

This problem can be modeled using the **binomial distribution** because we are dealing with a fixed number of trials (13 colleges) and each college either offers online courses or does not (binary outcome). The binomial distribution formula is:

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

Where:
- \( n = 13 \) (number of institutions)
- \( p = 0.96 \) (probability of an institution offering online courses)
- \( k \) is the number of institutions that offer online courses
- \( \binom{n}{k} \) is the binomial coefficient.

We are asked to find the probability that **at most 10** offer online courses, i.e., \( P(X \leq 10) \).

This can be computed as the cumulative probability:

\[
P(X \leq 10) = P(X = 0) + P(X = 1) + \dots + P(X = 10)
\]

Using the binomial probability formula for each term or through statistical software/calculators, we calculate this cumulative probability. Let me compute this for you.The probability that at most 10 out of 13 randomly selected institutions offer online courses is approximately \( 0.0135 \), or 1.35%.

Would you like more details about this calculation or have any questions?

Here are five related questions you might want to explore:
1. What is the probability that exactly 10 institutions offer online courses?
2. What is the probability that more than 10 institutions offer online courses?
3. What is the expected number of institutions that offer online courses?
4. How does the probability change if the proportion of colleges offering online courses is lower, say 90%?
5. What is the standard deviation of the number of institutions offering online courses?

**Tip:** The binomial distribution is ideal for modeling scenarios with a fixed number of independent trials, each having the same probability of success or failure.

More than 96 percent of the very largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 13 such institutions. We are interested in the number that offer distance courses. Find the probability that at most ten offer such courses.

Explore the probability calculation for at most 10 out of 13 colleges offering online courses, modeled using the binomial distribution with a 96% success rate. Learn about cumulative probability and binomial distribution formula applications in real-world scenarios. Suitable for college-level statistics students.

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