30% of all college students major in STEM (Science, Technology, Engineering, and Math). If 49 college students are randomly selected, find the probability that exactly 15 of them major in STEM.
.1188
Incorrect

Round to 4 decimal places.

To solve this problem, we can use the **binomial probability formula**, which is applicable because we have a fixed number of trials (49 students), two possible outcomes (STEM or not STEM), and a constant probability of success (30% or 0.30 for STEM).

The binomial probability formula is:

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

Where:
- \( n \) = 49 (number of trials, or students selected)
- \( k \) = 15 (number of successes, or students majoring in STEM)
- \( p \) = 0.30 (probability of success, i.e., the student majors in STEM)
- \( \binom{n}{k} \) = \( \frac{n!}{k!(n-k)!} \) is the binomial coefficient.

### Steps:
1. **Binomial coefficient** \(\binom{49}{15}\):
   \[
   \binom{49}{15} = \frac{49!}{15!(49-15)!}
   \]

2. **Substitute values** into the binomial probability formula:
   \[
   P(X = 15) = \binom{49}{15} \cdot (0.30)^{15} \cdot (0.70)^{34}
   \]

After calculating this step-by-step, rounding the final answer to 4 decimal places gives:

\[
P(X = 15) \approx 0.1477
\]

### The probability that exactly 15 of the 49 college students major in STEM is approximately **0.1477**.

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Would you like further details or have any questions?

Here are 5 related questions:
1. How would the probability change if the percentage of STEM majors were 40%?
2. What is the expected number of STEM majors in a sample of 49 students?
3. How do you calculate the probability of getting at least 15 STEM majors?
4. What is the variance and standard deviation for this binomial distribution?
5. How would this problem change if we selected 100 students instead of 49?

**Tip**: When working with binomial probabilities, it is helpful to remember that the sum of probabilities for all possible outcomes (from 0 to \(n\)) equals 1.

30% of all college students major in STEM (Science, Technology, Engineering, and Math). If 49 college students are randomly selected, find the probability that exactly 15 of them major in STEM. Round to 4 decimal places.

Learn how to calculate the probability that exactly 15 out of 49 randomly selected college students major in STEM using binomial probability. This problem demonstrates key statistical concepts such as binomial distribution and the application of the binomial probability formula. Suitable for high school seniors or college-level students.

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