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The problem is asking to find the probability that exactly 2 out of 10 randomly selected physics majors belong to an ethnic minority, given that \( \frac{1}{3} \) of the physics majors are from an ethnic minority.

This is a **binomial probability** problem where:
- The total number of trials (students selected) \( n = 10 \),
- The probability of success (selecting a student from an ethnic minority) \( p = \frac{1}{3} \),
- The number of successes (students from ethnic minorities) \( k = 2 \).

The binomial probability formula is:

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

Where:
- \( \binom{n}{k} \) is the binomial coefficient \( \frac{n!}{k!(n-k)!} \),
- \( p \) is the probability of success,
- \( 1 - p \) is the probability of failure,
- \( k \) is the number of successes (2 in this case),
- \( n \) is the number of trials (10 here).

### Step-by-Step Solution:

1. **Binomial coefficient** \( \binom{10}{2} \):

\[
\binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45
\]

2. **Substitute values into the binomial formula**:

\[
P(X = 2) = 45 \left(\frac{1}{3}\right)^2 \left(\frac{2}{3}\right)^8
\]

3. **Simplify**:

\[
P(X = 2) = 45 \times \frac{1}{9} \times \left(\frac{2}{3}\right)^8
\]
\[
= 45 \times \frac{1}{9} \times \frac{256}{6561}
\]
\[
= 45 \times \frac{256}{59049}
\]
\[
= \frac{11520}{59049}
\]
\[
\approx 0.195
\]

So, the probability that exactly 2 out of the 10 students belong to an ethnic minority is approximately **0.195**, or 19.5%.

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Would you like more details or have any questions?

Here are five related questions:
1. How is the binomial distribution used in real-world applications?
2. What is the expected value in a binomial distribution, and how is it calculated?
3. How does the probability change if the number of selected students increases?
4. Can this problem be solved using a normal approximation to the binomial distribution?
5. How does the variance of a binomial distribution depend on the parameters \( n \) and \( p \)?

**Tip:** The binomial distribution is ideal for problems involving repeated independent trials with two possible outcomes, like success or failure.

In a certain college, 1/3 of the physics majors belong to ethnic minorities. If 10 students are selected at random from the physics majors, what is the probability that exactly 2 belong to an ethnic minority?

Learn how to solve a binomial probability problem involving the selection of students from a minority group in a physics department. This problem focuses on calculating the probability that exactly 2 out of 10 randomly selected students belong to an ethnic minority, using the binomial probability formula. Suitable for Grades 10-12.

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