In a certain school, 70 percent of the student population is female. What is the probability that exactly 20 students are females, supposing 50 students are sampled? Choose one.

This is a **binomial probability problem**, where we want to find the probability that exactly 20 students are female in a sample of 50 students. Here's how we can break it down:

- The total number of trials (students sampled) is \( n = 50 \).
- The probability of success (selecting a female) is \( p = 0.7 \).
- The probability of failure (selecting a male) is \( q = 1 - p = 0.3 \).
- We want to find the probability that exactly \( k = 20 \) students are female.

The binomial probability formula is:

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

where:

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

Plugging in the values:

\[
P(X = 20) = \binom{50}{20} (0.7)^{20} (0.3)^{30}
\]

You would typically calculate this using a calculator or software, as the binomial coefficient \(\binom{50}{20}\) is large. The final probability would give the likelihood of exactly 20 females in the sample of 50 students.

If you're given multiple choice options, I can help verify which is closest to this value.

Would you like further details on the calculation or the use of a calculator? Here are related questions:

1. What is the probability that fewer than 20 students are females in the same scenario?
2. How does the result change if 100 students are sampled instead of 50?
3. Can the probability be approximated using a normal distribution?
4. How would the result change if the percentage of female students is 60% instead of 70%?
5. What is the variance and standard deviation for the number of female students sampled?

**Tip**: For large binomial calculations, using a normal approximation can simplify the process.

In a certain school, 70 percent of the student population is female. What is the probability that exactly 20 students are females, supposing 50 students are sampled?

Explore the probability of selecting exactly 20 female students from a sample of 50, where 70% of the student population is female. Learn how to use the binomial probability formula for this calculation. Ideal for Grades 10-12, this problem covers probability concepts and the binomial distribution.

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