Question

25% of all college students major in stem (science, technology, engineering, and math). if 50 college students are randomly selected, find the probability that

a. exactly 13 of them major in stem.

b. at most 16 of them major in stem.

c. at least 14 of them major in stem.

d. between 9 and 13 (including 9 and 13) of them major in stem.

Explanation:

This is a binomial probability problem where \(n = 50\) (number of trials, i.e., number of students selected), \(p=0.25\) (probability of success, i.e., probability that a student majors in STEM). The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\).

Step1: Calculate binomial coefficient for part a

For \(n = 50\), \(k = 13\), \(C(50,13)=\frac{50!}{13!(50 - 13)!}=\frac{50!}{13!37!}\)
\[C(50,13)=\frac{50\times49\times\cdots\times38}{13\times12\times\cdots\times1}\]
\[P(X = 13)=C(50,13)\times(0.25)^{13}\times(0.75)^{37}\]
\[P(X = 13)\approx0.130\]

Step2: Calculate cumulative - probability for part b

We use the cumulative - binomial probability formula \(P(X\leq k)=\sum_{i = 0}^{k}C(n,i)\times p^{i}\times(1 - p)^{n - i}\)
For \(k = 16\), \(P(X\leq16)=\sum_{i = 0}^{16}C(50,i)\times(0.25)^{i}\times(0.75)^{50 - i}\approx0.786\)

Step3: Calculate complementary probability for part c

\(P(X\geq14)=1 - P(X\leq13)\)
First, \(P(X\leq13)=\sum_{i = 0}^{13}C(50,i)\times(0.25)^{i}\times(0.75)^{50 - i}\)
\(P(X\leq13)\approx0.443\)
\(P(X\geq14)=1 - 0.443 = 0.557\)

Step4: Calculate cumulative - probability difference for part d

\(P(9\leq X\leq13)=P(X\leq13)-P(X\leq8)\)
\(P(X\leq8)=\sum_{i = 0}^{8}C(50,i)\times(0.25)^{i}\times(0.75)^{50 - i}\approx0.102\)
\(P(9\leq X\leq13)=0.443-0.102 = 0.341\)

Answer:

a. Approximately \(0.130\)
b. Approximately \(0.786\)
c. Approximately \(0.557\)
d. Approximately \(0.341\)