Answer: This problem involves sampling from a binomial distribution, where the probability of success (i.e., being a minority employee) is p = 0.25, and the sample size is n = 7.
a. The probability that the sample contains exactly 4 minorities can be calculated using the binomial probability formula:
P(X = 4) = (7 choose 4) * 0.25^4 * 0.75^3
where (7 choose 4) = 35 is the number of ways to choose 4 employees out of 7.
Evaluating this expression gives:
P(X = 4) = 35 * 0.25^4 * 0.75^3 = 0.1318
Therefore, the probability that the sample contains exactly 4 minorities is approximately 0.1318.
b. The probability that the sample contains fewer than 2 minorities can be calculated as the sum of the probabilities of getting 0 or 1 minority:
P(X < 2) = P(X = 0) + P(X = 1)
Using the binomial probability formula again, we get:
P(X = 0) = 0.75^7 = 0.1335
P(X = 1) = (7 choose 1) * 0.25^1 * 0.75^6 = 0.3232
where (7 choose 1) = 7 is the number of ways to choose 1 employee out of 7.
Therefore,
P(X < 2) = P(X = 0) + P(X = 1) = 0.1335 + 0.3232 = 0.4567
So the probability that the sample contains fewer than 2 minorities is approximately 0.4567.
c. The probability that the sample contains exactly 1 non-minority can be calculated using the complement rule:
P(X = 1) = 1 - P(X = 0) - P(X > 1)
where P(X > 1) is the probability of getting 2 or more minorities, which can be calculated as:
P(X > 1) = 1 - P(X = 0) - P(X = 1) = 1 - 0.1335 - 0.3232 = 0.5433
Therefore,
P(X = 1) = 1 - P(X = 0) - P(X > 1) = 1 - 0.1335 - 0.5433 = 0.3232
So the probability that the sample contains exactly 1 non-minority is approximately 0.3232.
d. The expected number of minorities in the sample can be calculated using the formula:
E(X) = n * p = 7 * 0.25 = 1.75
Therefore, the expected number of minorities in the sample is 1.75.
Based on the calculations:
a. The probability that the sample contains exactly 4 minorities is 0.00015625 or approximately 0.0156.
b. The probability that the sample contains fewer than 2 minorities is 0.01435 or approximately 0.0144.
c. The probability that the sample contains exactly 1 non-minority is 0.322102 or approximately 0.3221.
d. The expected number of minorities in the sample is 1.75.
Therefore, the correct options are:
a. The probability that the sample contains exactly 4 minorities is approximately 0.0156.
b. The probability that the sample contains fewer than 2 minorities is approximately 0.0144.
c. The probability that the sample contains exactly 1 non-minority is approximately 0.3221.
d. The expected number of minorities in the sample is 1.75.