Chapter 8.6, Problem 63E

(a).

To determine

To find:

The probability that the all six contacts.

Answer to Problem 63E

The required probability is $0.000064$ .

Explanation of Solution

Given:

A sales representative makes sales on approximately one-fifth of all calls. On a given day, the representative contacts six potential clients.

Concept Used:

Apply the formula of the probability:

  $\begin{array}{c}P\left(A\right)=\frac{NumberofFavourableOutcome}{TotalNumberofFavourableOutcome}\\ =\frac{n\left(A\right)}{n\left(S\right)}\end{array}$

Calculation:

The probability of a single call being a success is $\frac{1}{5}$ .

  $\begin{array}{c}P\left(allsixcontacts\right)={\left(\frac{1}{5}\right)}^6\\ =\frac{1}{15625}\\ =0.000064\end{array}$

Conclusion:

Hence, the required probability is $0.000064$ .

(b).

To determine

To find:

The probability that none of the contacts.

Answer to Problem 63E

The required probability is $0.262144$ .

Explanation of Solution

Given:

A sales representative makes sales on approximately one-fifth of all calls. On a given day, the representative contacts six potential clients.

Concept Used:

Apply the formula of the probability:

  $\begin{array}{c}P\left(A\right)=\frac{NumberofFavourableOutcome}{TotalNumberofFavourableOutcome}\\ =\frac{n\left(A\right)}{n\left(S\right)}\end{array}$

Calculation:

The probability of a single call being a success is $\frac{1}{5}$ .

The probability of a single call being a unsuccess is $1-\frac{1}{5}=\frac{4}{5}$ .

  $\begin{array}{c}P\left(failallsixcontacts\right)={\left(\frac{4}{5}\right)}^6\\ =\frac{4096}{15625}\\ =0.262144\end{array}$

Conclusion:

Hence, the required probability is $0.262144$ .

(c).

To determine

To find:

The probability that at least one contact.

Answer to Problem 63E

The required probability is $0.737856$ .

Explanation of Solution

Given:

A sales representative makes sales on approximately one-fifth of all calls. On a given day, the representative contacts six potential clients.

Concept Used:

Apply the formula of the probability:

  $\begin{array}{c}P\left(A\right)=\frac{NumberofFavourableOutcome}{TotalNumberofFavourableOutcome}\\ =\frac{n\left(A\right)}{n\left(S\right)}\end{array}$

Calculation:

The probability of a single call being a success is $\frac{1}{5}$ .

The probability of a single call being a unsuccess is $1-\frac{1}{5}=\frac{4}{5}$ .

  $\begin{array}{c}P\left(failallsixcontacts\right)={\left(\frac{4}{5}\right)}^6\\ =\frac{4096}{15625}\\ =0.262144\end{array}$

  $\begin{array}{c}P\left(atleastonecontact\right)=1-{\left(\frac{4}{5}\right)}^6\\ =1-0.262144\\ =0.737856\end{array}$

Conclusion:

Hence, the required probability is $0.737856$ .