Question

About $32\%$ of students participate in a community volunteer program outside of school. If $30$students are selected at random, find the probability that at most $14$ of them participate in a community volunteer program outside of school. Use the $TI-83+orTI-84$ calculator to find the answer.

Short answer

The required probability that at most $14$students out of $30$students selected at random participate in a community volunteer programme outside of school is$96.95\%or0.9695$

Step-by-step solution

Content Introduction

Let the random variable $X=$number of students participate in community volunteer program outside of school out of$30$students selected at random.

Content Explanation

Assuming that $X=0,1,2,.........30$

The total number of students is n = $30$

The randomly selected students to participate in a community volunteer program outside of school is p =$0.32$

Now,

$$\begin{gathered} q=1-p \\ q=1-0.32 \\ q=0.68 \end{gathered}$$

Then,$X$follows a binomial distribution

Therefore, the required probability is

$P(X\le 14)=\sum _{x=0}^{14} P(X=x)$

Substituting the values we get,

$\sum _{x=0}^{14}{ }^n{C}_x{p}^x{q}^{n-x}=\sum ^n C_x\left(0.32{)}^x\right(0.68{)}^{30-x}$

Upon expanding the summation, we get

${}^{30}{C}_0\left(0.32{)}^0\right(0.68{)}^{30-0}{+}^{30}{C}_1\left(0.32{)}^1\right(0.68{)}^{30-1}{+}^{30}{C}_2\left(0.32{)}^2\right(0.68{)}^{30-2}+\dots {\dots }^{30}{C}_{14}\left(0.32{)}^{14}\right(0.68{)}^{30-14}$

$$\begin{gathered} =0.9695 \\ =96.5\% \end{gathered}$$