36% of college students say they use credit cards because of the rewards program. You randomly select 10 college students and ask each to name the reason he or she uses credit cards. Find the probability that the number of college students who say they use credit cards because of the rewards program is

(a) exactly two,
(b) more than two, and
(c) between two and five inclusive.

Solution

Here $X$ denote the number of students who use credit cards because of the rewards program out of selected 10 college students.

$p$ be the percent of the students who use credit cards because of the rewards program.

Given that $p=0.36$ and $n =10$. Thus$X\sim B(10, 0.36)$.

The probability mass function of $X$ is
$$ \begin{aligned} P(X=x) &= \binom{10}{x} (0.36)^x (1-0.36)^{10-x}, \\ &\qquad \; x=0,1,\cdots, 10. \end{aligned} $$

a. The probability that exactly 2 college students who say they use credit cards because of the rewards program is

$$ \begin{aligned} P(X=2) & =\binom{10}{2} (0.36)^{2} (1-0.36)^{10-2}\\ &=0.1642 \end{aligned} $$

b. The probability that more than 2 college students who say they use credit cards because of the rewards program is

$$ \begin{aligned} P(X > 2) & =1- P(X\leq 2)\\ &= 1- \sum_{x=0}^{2} P(x)\\ & =1- \bigg(P(X=0)+P(X=1)+P(X=2)\bigg)\\ &= 1-\bigg(0.0115+0.0649+0.1642\bigg) \\ &= 1-0.2405\\ &= 0.7595 \end{aligned} $$

c. The probability that between 2 and 5 (inclusive) college students who say they use credit cards because of the rewards program is

$$ \begin{aligned} P(2\leq X \leq 5) & = \bigg(P(X=2)+P(X=3)+P(X=4)+P(X=5)\bigg)\\ &= \bigg(0.1642+0.2462+0.2424+0.1636\bigg) \\ &= 0.8164 \end{aligned} $$