Chapter 8.3, Problem 8.59E

(a)

To determine

To show that the four requirements for a binomial setting are met.

Explanation of Solution

The four conditions for a binomial setting are: binary (success/failure), independent trials, fixed number of trials and the probability of success is the same for each trial.

Binary: first/not first.

Independent: Yes, the chance of somebody greeting the dog first is not dependent on another person.

Number: $12$

Success: $66\%$

Thus the setting is binomial.

(b)

To determine

To calculate the probability that exactly $7\text{ of the 12}$ dog owners greet their dog first when they arrive home.

Answer to Problem 8.59E

  $0.1963$ .

Explanation of Solution

Given:

  $\begin{array}{l}n=12,k=7\\ p=0.66\end{array}$

  $\begin{array}{l}$ P(X=7)=C{}_7\times {(0.66)}^7\times {(1-0.66)}^{12-7}$=792\times {0.66}^7\times {0.34}^5\\ =0.1963=19.63\%\end{array}$

(c)

To determine

To calculate the probability that at least $\text{5 of the 12}$ dog owners greet their dog first when they arrive home.

Answer to Problem 8.59E

  $0.0521$ .

Explanation of Solution

Given:

  $\begin{array}{l}n=12,k=5\\ p=0.66\end{array}$

  $\begin{array}{l}$ P(X=5)=C{}_5\times {(0.66)}^5\times {(1-0.66)}^{12-5}$=792\times {0.66}^5\times {0.34}^7\\ =0.0521=5.21\%\end{array}$