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%.

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.