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:

Step1: Identify the binomial distribution parameters

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

Step2: Calculate probability for part a

For $k = 13$, $C(50,13)=\frac{50!}{13!(50 - 13)!}=\frac{50!}{13!37!}$
$P(X = 13)=C(50,13)\times(0.25)^{13}\times(0.75)^{37}$
$C(50,13)=\frac{50!}{13!37!}=\frac{50\times49\times\cdots\times38}{13!}\approx1.589\times 10^{12}$
$P(X = 13)\approx1.589\times 10^{12}\times(0.25)^{13}\times(0.75)^{37}\approx0.133$

Step3: Calculate probability for part b

$P(X\leq16)=\sum_{k = 0}^{16}C(50,k)\times(0.25)^{k}\times(0.75)^{50 - k}$
Using a binomial - cumulative distribution function (e.g., in a calculator or software like R: pbinom(16,50,0.25)), $P(X\leq16)\approx0.898$

Step4: Calculate probability for part c

$P(X\geq14)=1 - P(X\leq13)$
Using the binomial - cumulative distribution function, $P(X\leq13)=\sum_{k = 0}^{13}C(50,k)\times(0.25)^{k}\times(0.75)^{50 - k}$
In R: pbinom(13,50,0.25), $P(X\leq13)\approx0.446$
$P(X\geq14)=1 - 0.446 = 0.554$

Step5: Calculate probability for part d

$P(9\leq X\leq13)=\sum_{k = 9}^{13}C(50,k)\times(0.25)^{k}\times(0.75)^{50 - k}$
$P(9\leq X\leq13)=P(X\leq13)-P(X\leq8)$
In R: pbinom(13,50,0.25)-pbinom(8,50,0.25)
$P(X\leq8)=\sum_{k = 0}^{8}C(50,k)\times(0.25)^{k}\times(0.75)^{50 - k}\approx0.104$
$P(9\leq X\leq13)\approx0.446-0.104 = 0.342$

Answer:

a. $0.133$
b. $0.898$
c. $0.554$
d. $0.342$