Question

In a large university, 20% of the students are business majors. A random sample of 100 students is selected, and their majors are recorded.

d. What is the probability that the sample contains between 12 and 14 business majors?

Answer 1

Let X be the number of students that are business majors.

Here n = 100, p = 0.2

Hence, X ~ Binomial(n = 100, p = 0.2)

$P(X = x) = \binom{n}{x}\times p^x \times (1-p)^{n-x}$

$P(12 < x<14) = \binom{100}{12}\times (0.2)^{12} \times (1-0.2)^{100-12} + \binom{100}{13}\times (0.2)^{13} \times (1-0.2)^{100-13} + \binom{100}{14}\times (0.2)^{14} \times (1-0.2)^{100-14}$$P(12 < x<14) \approx 0.06786$

Hence, the probability that the sample contains between 12 and 14 business majors is 0.06786