Chapter 11.4, Problem 16PPS

To determine

What is the probability that $3$ out of $15$ random high school upperclassmen think this way?

Answer to Problem 16PPS

  $\boxed{22.52\%}$

Explanation of Solution

Given information:

According to a recent survey, $25\%$ of high school upperclassmen think that the junior-senior prom is the most important event of the school year. What is the probability that $3$ out of $15$ random high school upperclassmen think this way?

Calculation:

Consider a survey in which it is found that $25\%$ of the high school upperclassmen think that the junior-senior prom is the most important event of the school year.

The probability of successes in $n$ trials of a binomial experiment is,

  $P(k)=C{}_k{(p)}^k{(q)}^{n-k}$

Here, $p$ represents the probability of success and $q$ represents the probability of failure which is equal to,

  $q=1-p$

Since, $25\%$ of the high school upperclassmen think that the junior-senior prom is the most important event of the school year so,

  $p=25\%$

  $=0.25$

The probability of failure is,

  $q=1-p$

  $\begin{array}{l}=1-0.25\\ =0.75\end{array}$

Substitute $p=0.25,q=0.75,k=3,n=15$ in the above formula to find the probability that $3$ out of $15$ random high school upperclassmen think that the junior-senior prom is the most important event of the school year

  $P(3)=C{}_3{(0.25)}^3{(0.75)}^{15-3}$

  $\begin{array}{l}=455{(}^{0.25}{(}^{0.75}\\ =0.2252\\ =22.52\%\end{array}$

Hence, the probability that $3$ out of $15$ random high school upperclassmen think that the junior-senior prom is the most important event of the school year is $\boxed{22.52\%}$ .