91% of the largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 43 such intitutions, and let X be the number of intitutions that offer distance learning courses.
What is the distribution of X? X ~  (,)
Please show the following answers to 4 decimal places.
What is the probability that exactly 37 intitutions in this study offer distance learning courses? 
What is the probability that more than 37 intitutions in this study offer distance learning courses? 
What is the probability that at most 37 intitutions in this study offer distance learning courses? 
What is the probability that between 35 and 38 (including 35 and 38) intitutions in this study offer distance learning courses? 

Given that 91% of the largest colleges and universities offer online courses and that we randomly pick 43 such institutions, we can model the number of institutions that offer distance learning courses using a **binomial distribution**.

The binomial distribution is described by:

\[
X \sim \text{Binomial}(n, p)
\]

where:
- \( n = 43 \) (the number of institutions),
- \( p = 0.91 \) (the probability that a randomly selected institution offers distance learning).

Thus, \( X \sim \text{Binomial}(43, 0.91) \).

### 1. **Probability that exactly 37 institutions offer distance learning courses**
This is a binomial probability calculated as:

\[
P(X = 37) = \binom{43}{37} p^{37} (1-p)^{43-37}
\]

I will calculate this.

### 2. **Probability that more than 37 institutions offer distance learning courses**
This is the sum of probabilities from 38 to 43 institutions:

\[
P(X > 37) = P(X = 38) + P(X = 39) + P(X = 40) + P(X = 41) + P(X = 42) + P(X = 43)
\]

I will compute this as well.

### 3. **Probability that at most 37 institutions offer distance learning courses**
This is the sum of probabilities from 0 to 37 institutions:

\[
P(X \leq 37) = \sum_{k=0}^{37} P(X = k)
\]

### 4. **Probability that between 35 and 38 (including 35 and 38) institutions offer distance learning courses**
This is the sum of probabilities from 35 to 38:

\[
P(35 \leq X \leq 38) = P(X = 35) + P(X = 36) + P(X = 37) + P(X = 38)
\]

Now, I will compute each of these probabilities.

91% of the largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 43 such institutions, and let X be the number of institutions that offer distance learning courses. What is the distribution of X? X ~ (,). Please show the following answers to 4 decimal places. What is the probability that exactly 37 institutions in this study offer distance learning courses? What is the probability that more than 37 institutions in this study offer distance learning courses? What is the probability that at most 37 institutions in this study offer distance learning courses? What is the probability that between 35 and 38 (including 35 and 38) institutions in this study offer distance learning courses?

Explore the binomial distribution for a study of 43 colleges where 91% offer online courses. Learn how to calculate the probability for exactly 37, more than 37, at most 37, and between 35 and 38 institutions offering distance learning. Suitable for undergraduate students.

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