31% of all college students major in STEM (Science, Technology, Engineering, and Math). If 32 college students are randomly selected, find the probability that exactly 10 of them major in STEM. 
Round to 4 decimal places.

This problem involves binomial probability, where we are given:

- The probability of success (majoring in STEM) \( p = 0.31 \)
- The probability of failure (not majoring in STEM) \( q = 1 - p = 0.69 \)
- The number of trials \( n = 32 \) (since we are selecting 32 students)
- The number of successes \( x = 10 \)

The binomial probability formula is:

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

Where:
- \( \binom{n}{x} \) is the binomial coefficient, representing the number of ways to choose \( x \) successes from \( n \) trials, calculated as:

\[
\binom{n}{x} = \frac{n!}{x!(n-x)!}
\]

Let's calculate this step by step:

1. \( n = 32 \), \( x = 10 \), \( p = 0.31 \), and \( q = 0.69 \)
2. The binomial coefficient \( \binom{32}{10} \) is:

\[
\binom{32}{10} = \frac{32!}{10!(32-10)!} = \frac{32!}{10!22!}
\]

3. The probability is then:

\[
P(X = 10) = \binom{32}{10} (0.31)^{10} (0.69)^{22}
\]

Let me compute this for you.The probability that exactly 10 out of 32 randomly selected college students major in STEM is approximately \( 0.1506 \) (rounded to four decimal places).

Would you like more details on any part of the process, or do you have any other questions?

Here are 5 related questions to consider:
1. What is the probability that fewer than 10 students major in STEM?
2. How does the probability change if the number of selected students increases?
3. What if 40% of students major in STEM—how would the probability change?
4. What is the expected number of students majoring in STEM out of 32?
5. What is the probability that between 8 and 12 students major in STEM?

**Tip**: For binomial distributions, the mean is \( n \times p \) and the variance is \( n \times p \times (1 - p) \).

31% of all college students major in STEM (Science, Technology, Engineering, and Math). If 32 college students are randomly selected, find the probability that exactly 10 of them major in STEM. Round to 4 decimal places.

Solve a binomial probability problem where 31% of students major in STEM. Learn how to calculate the probability that exactly 10 out of 32 randomly selected students major in STEM using the binomial distribution formula. Suitable for college-level statistics students.

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