Chapter 11.4, Problem 15PPS

To determine

What is the probability that $10$ out of $12$ random high school students drive their own car?

Answer to Problem 15PPS

  $\boxed{18.35\%}$

Explanation of Solution

Given information:

According to a recent survey, $92\%$ of high school seniors drive their own car. What is the probability that $10$ out of $12$ random high school students drive their own car?Calculation:

Consider a survey in which it is found that $92\%$ of the high school students drive their own car.

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, $92\%$ of the high school students drive their own car. so,

  $p=92\%$

  $=0.92$

The probability of failure is,

  $q=1-p$

  $\begin{array}{l}=1-0.92\\ =0.08\end{array}$

Substitute $p=0.92,q=0.08,k=10,n=12$ in the above formula to find the probability that $10$ out of $12$ random high school students drive their own car

  $P(6)=C{}_{10}{(0.92)}^{10}{(0.08)}^{12-10}$

  $\begin{array}{l}=66{(}^{0.92}{(}^{0.08}\\ =0.1835\\ =18.35\%\end{array}$

Hence, the probability that $10$ out of $12$ random high school students drive their own car is $\boxed{18.35\%}$ .