QuestionIn a certain state, $35.7 \%$ of all community college students belong to ethnic minorities. Find the probabilities of the following results in a random sample of 12 of the community college students. a. Exactly 3 belong to an ethnic minority. b. Three or fewer belong to an ethnic minonity. c. Exactly 7 do not belong to an ethnic minority. d. Eight or more do not belong to an ethnic minority.

Studdy Solution
Calculate the probability of eight or more students not belonging to an ethnic minority. This is equivalent to four or fewer students belonging to an ethnic minority:
$P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)$
Compute each probability using the binomial formula and sum them:
$P(X = k) = \binom{12}{k} (0.357)^k (0.643)^{12-k} \quad \text{for } k = 0, 1, 2, 3, 4$
The probabilities for each part are calculated using the binomial distribution formula. Use a calculator or statistical software to compute the exact numerical values for each probability.