Chapter 11.4, Problem 14PPS

To determine

What is the probability that $6$ out of $10$ random high school students own a personal media player?

Answer to Problem 14PPS

  $\boxed{4.01\%}$

Explanation of Solution

Given information:

According to a recent survey, $85\%$ of high school students own a personal media player. What is the probability that $6$ out of $10$ random high school students own a personal media player?Calculation:

Consider a survey in which it is found that $85\%$ of the high school students own a personal media player.

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 $85\%$ of the students own a personal media player, so,

  $p=85\%$

  $=0.85$

The probability of failure is,

  $q=1-p$

  $\begin{array}{l}=1-0.85\\ =0.15\end{array}$

Substitute $p=0.85,q=0.15,k=6,n=10$ in the above formula to find the probability that $6$ out of $10$ random high school students own a personal media player.

  $P(6)=C{}_6{(0.85)}^6{(0.15)}^{10-6}$

  $\begin{array}{l}=210{(}^{0.85}{(}^{0.15}\\ =0.0401\\ =4.01\%\end{array}$

Hence, the probability that $6$ out of $10$ random high school students own a personal media player is $\boxed{4.01\%}$ .