Chapter 13, Problem 17STP

a.

To determine

To find the probabilities associated with the students that do not buy compact discs.

Explanation of Solution

Given:

A survey that found that $91\%$ of the high school students do not buy compact discs and $8$ random students are chosen from this population.

Calculation:

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

  $P(k)=C(n,k)p^k{q}^{n-k}$

Here p represents the probability of success and q represents the probability of failure.

Since $91\%$ of the students do not buy compact discs, so,

  $p=0.91$

While the remaining 9% buy them, so,

  $\begin{array}{l}q=1-0.91\\ q=0.09\end{array}$

Substitute in the above formula to find the probability that none of the $8$ random students surveyed do not buy compact discs,

  $P(0)=C(8,0){(0.91)}^0{(0.09)}^{10}=0.0000$

Thus, the probability that none of the 8 random students do not buy compact discs is.

Substitute in the above formula to find the probability that exactly 1 of the 8 random students surveyed do not buy compact discs,

  $P(1)=C(8,1){(0.91)}^1{(0.09)}^9=0.0000$

Substitute in the above formula to find the probability that exactly 2 of the 8 random students surveyed do not buy compact discs,

  $P(2)=C(8,2){(0.91)}^2{(0.09)}^8=0.0000$

......

Thus, the probability that all 8 random students surveyed do not buy compact discs is.

  $P(8)=C(8,8){(0.91)}^8{(0.09)}^0=0.4703$

b.

To determine

To find the probability that at least $7$ out of $8$ random students surveyed do not buy compact discs.

Answer to Problem 17STP

  $0.8424$

Explanation of Solution

Given:

A survey that found that $91\%$ of the high school students do not buy compact discs and $8$ random students are chosen from this population.

Calculation:

The number of students out of $8$ random students who do not buy compact discs could be $7$ or $8$ .

From part a, it is clear that,

  $\begin{array}{l}P(7)=0.3721\\ P(8)=0.4703\end{array}$

So, the probability that at least $7$ out of $8$ random students surveyed do not buy compact discs is,

  $\begin{array}{l}P(\text{at least }7)=P(7)+P(8)\\ =0.3721+0.4703\\ =0.8424\end{array}$

Thus, the probability that at least $7$ out of $8$ random students surveyed do not buy compact discs is $0.8424$ .

c.

To determine

To find the probabilities associated with the students that do not buy compact discs.

Answer to Problem 17STP

  $7$

Explanation of Solution

Given:

A survey that found that $91\%$ of the high school students do not buy compact discs and $8$ random students are chosen from this population.

Calculation:

In general, the mean, of any binomial distribution is,

  $\mu =np$

Here n is the number of trials and p is the probability of success.

For the above situation, it is clear that $n=8,p=0.91$ . Substitute these values in the above formula to find the number of students expected who do not buy compact discs,

  $\begin{array}{l}\mu =np\\ \mu =8\times 0.91\\ \mu =7.28\\ \mu \approx 7\end{array}$

Thus, the number of students expected who do not buy compact discs is $7$ .