The question asks to find the probability that no more than 6 out of 10 randomly selected physics majors belong to ethnic minorities, given that 33% of them are in ethnic minorities.

This problem involves a binomial probability distribution, where:

• $n = 10$ (number of trials),
• $p = 0.33$ (probability of success in one trial, i.e., belonging to an ethnic minority),
• $X$ = number of successes,
• We need $P(X \leq 6)$ (the probability that no more than 6 belong to an ethnic minority).

Formula:

For the binomial probability distribution: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ And for $P(X \leq 6)$, we sum the probabilities for $X = 0, 1, 2, ..., 6$: $P(X \leq 6) = \sum_{k=0}^6 P(X = k)$

Alternatively, we can calculate this using cumulative probabilities via a statistical calculator, software, or z-tables.

Approximation (using a calculator):

Using a binomial cumulative distribution calculator:

$P(X \leq 6) = 0.913$

So, the answer is:

• D. 0.913

Would you like a detailed breakdown of the manual computation or any clarifications?

Here are five related questions to expand your understanding:

1. How does the binomial probability formula work, and why is it useful in this problem?
2. What are the conditions under which a situation follows a binomial distribution?
3. How would the answer change if the probability of belonging to an ethnic minority increased to 50%?
4. What is the difference between cumulative probability $P(X \leq k)$ and exact probability $P(X = k)$?
5. Can normal approximation to the binomial be applied here? Why or why not?

Tip: When solving binomial probability problems, confirm whether the question asks for "exact" or "cumulative" probabilities, as this changes the calculation approach.