A history class is comprised of 4 female and 5 male students. If the instructor of the class randomly chooses 5 students from the class for an oral exam, what is the probability that 3 female students and 2 male students will be selected? Round your answer to 3 decimal places.

To solve the problem, we can use the concept of combinations and the probability formula.

### Step 1: Calculate the total number of ways to choose 5 students out of 9.
The total number of students in the class is \(4 + 5 = 9\). The number of ways to choose 5 students out of 9 is given by:

\[
\text{Total combinations} = \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9 \times 8 \times 7 \times 6 \times 5}{5 \times 4 \times 3 \times 2 \times 1} = 126
\]

### Step 2: Calculate the number of favorable outcomes (selecting 3 females and 2 males).
- The number of ways to choose 3 female students out of 4 is:

\[
\binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4 \times 3 \times 2}{3 \times 2 \times 1} = 4
\]

- The number of ways to choose 2 male students out of 5 is:

\[
\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
\]

- The number of favorable outcomes is the product of these two combinations:

\[
\text{Favorable combinations} = \binom{4}{3} \times \binom{5}{2} = 4 \times 10 = 40
\]

### Step 3: Calculate the probability.
The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes:

\[
P(\text{3 females and 2 males}) = \frac{\text{Favorable combinations}}{\text{Total combinations}} = \frac{40}{126} \approx 0.317
\]

### Final Answer:
The probability that 3 female students and 2 male students will be selected is approximately **0.317**.

Do you want details or have any questions?

Here are some related questions:

1. How would the probability change if there were 6 male students instead of 5?
2. What is the probability of selecting exactly 4 female students and 1 male student?
3. How many ways can you select all 4 female students from the class?
4. What is the probability of selecting all male students?
5. How does the binomial coefficient \(\binom{n}{k}\) work?
6. What is the complement of the event that 3 female and 2 male students are selected?
7. How does the concept of combinations differ from permutations in probability?
8. Can this problem be solved using the hypergeometric distribution?

**Tip:** In probability problems involving selection, always differentiate between combinations and permutations, as they lead to different results depending on whether order matters.

Calculate the probability of selecting 3 female students and 2 male students from a history class of 4 females and 5 males. Learn the concept of combinations and apply the binomial coefficient formula to solve the problem step-by-step. Suitable for high school students learning combinatorics and probability.

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