26% of all college students major in STEM (Science, Technology, Engineering, and Math). If 35 college students are randomly selected, find the probability that a. Exactly 10 of them major in STEM. b. At most 11 of them major in STEM. c. At least 7 of them major in STEM. d. Between 7 and 11 (including 7 and 11) of them major in STEM.

This problem involves the **binomial distribution** since it deals with a fixed number of trials (35 college students), two possible outcomes (majoring in STEM or not), and a constant probability of success (26%).

The binomial distribution formula is:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
Where:
- \(n\) = number of trials (students selected) = 35,
- \(k\) = number of successes (students majoring in STEM),
- \(p\) = probability of success (STEM major) = 0.26,
- \(\binom{n}{k}\) is the binomial coefficient.

Let's solve each part.

---

### a. **Exactly 10 of them major in STEM**
We want the probability that exactly 10 out of 35 students major in STEM, i.e., \( P(X = 10) \).

\[
P(X = 10) = \binom{35}{10} (0.26)^{10} (0.74)^{25}
\]

---

### b. **At most 11 of them major in STEM**
Here, we need to find the probability that 11 or fewer students major in STEM, i.e., \( P(X \leq 11) \).

This is the sum of probabilities for 0 to 11 students:
\[
P(X \leq 11) = \sum_{k=0}^{11} P(X = k)
\]

---

### c. **At least 7 of them major in STEM**
We want the probability that 7 or more students major in STEM, i.e., \( P(X \geq 7) \).

This is equivalent to:
\[
P(X \geq 7) = 1 - P(X \leq 6)
\]

---

### d. **Between 7 and 11 (including 7 and 11) major in STEM**
This means the probability that the number of STEM majors is between 7 and 11, inclusive, i.e., \( P(7 \leq X \leq 11) \).

This is the sum of probabilities from 7 to 11:
\[
P(7 \leq X \leq 11) = \sum_{k=7}^{11} P(X = k)
\]

---

I will now compute these probabilities.Here are the probabilities for each part:

### a. **Exactly 10 of them major in STEM**:  
\( P(X = 10) \approx 0.1394 \)

### b. **At most 11 of them major in STEM**:  
\( P(X \leq 11) \approx 0.8238 \)

### c. **At least 7 of them major in STEM**:  
\( P(X \geq 7) \approx 0.8421 \)

### d. **Between 7 and 11 (inclusive) major in STEM**:  
\( P(7 \leq X \leq 11) \approx 0.6659 \)

Do you have any further questions or need more details?

Here are 5 related questions:
1. How does changing the sample size to 50 affect the probabilities?
2. What is the expected number of students majoring in STEM?
3. How would the probabilities change if the percentage of STEM majors increased to 30%?
4. What is the probability that none of the 35 students major in STEM?
5. How can this binomial problem be approximated using a normal distribution?

**Tip:** Binomial distributions are best used when you have a fixed number of trials with two outcomes (success or failure) and a constant probability of success.

Solve a binomial probability problem involving STEM majors, using real-world data. 26% of students major in STEM. Learn to calculate the probability that exactly 10, at most 11, and at least 7 students major in STEM when 35 are randomly selected. A comprehensive guide for college-level statistics.

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