Chapter 13.5, Problem 13CFU

a.

To determine

To calculate: The probability that bill accepted by changer is legal.

Answer to Problem 13CFU

The probability that bill accepted by changer is legal is $\frac{69}{70}$ .

Explanation of Solution

Given information:

100 $1- bills were tested and 25 of the bills were counterfeit.

$\begin{array}{ccc}\text{Bill}& \text{Accepted}& \text{Rejected}\\ \text{Legal}& 69& 6\\ \text{Counterfeit }& 1& 24\end{array}$

Formula used:

Probability of an event E is $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Given an event Q, the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

Calculation:

Consider the provided information that 100 $1- bills were tested and 25 of the bills were counterfeit.

$\begin{array}{ccc}\text{Bill}& \text{Accepted}& \text{Rejected}\\ \text{Legal}& 69& 6\\ \text{Counterfeit }& 1& 24\end{array}$

Total number of bills are 100.

Let A be an event that bill is legal and B be an event that bill is accepted.

Recall that probability of an event E is $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Probability that bill is accepted is,

  $\begin{array}{c}P\left(B\right)=\frac{69+1}{100}\\ =\frac{70}{100}\end{array}$

Probability that bill is legal and bill is accepted is,

  $P\left(A\text{ and B}\right)=\frac{69}{100}$

Recall that given an event Q, the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

The probability that bill accepted by changer is legal.

  $\begin{array}{c}P\left(A|B\right)=\frac{\frac{69}{100}}{\frac{70}{100}}\\ =\frac{69}{70}\end{array}$

Thus, the probability that bill accepted by changer is legal is $\frac{69}{70}$ .

b.

To determine

To calculate: The probability that bill is rejected provided that bill is legal.

Answer to Problem 13CFU

The probability that bill is rejected provided that bill is legal is $\frac{2}{25}$ .

Explanation of Solution

Given information:

100 $1- bills were tested and 25 of the bills were counterfeit.

$\begin{array}{ccc}\text{Bill}& \text{Accepted}& \text{Rejected}\\ \text{Legal}& 69& 6\\ \text{Counterfeit }& 1& 24\end{array}$

Formula used:

Probability of an event E is $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Given an event Q, the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

Calculation:

Consider the provided information that 100 $1- bills were tested and 25 of the bills were counterfeit.

$\begin{array}{ccc}\text{Bill}& \text{Accepted}& \text{Rejected}\\ \text{Legal}& 69& 6\\ \text{Counterfeit }& 1& 24\end{array}$

Total number of bills are 100.

Let A be an event that bill is rejected and B be an event that bill is legal.

Recall that probability of an event E is $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Probability that bill is legal,

  $\begin{array}{c}P\left(B\right)=\frac{69+6}{100}\\ =\frac{75}{100}\end{array}$

Probability that bill is rejected and bill is legal is,

  $P\left(A\text{ and B}\right)=\frac{6}{100}$

Recall that given an event Q, the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

The probability that bill is rejected provided that bill is legal.

  $\begin{array}{c}P\left(A|B\right)=\frac{\frac{6}{100}}{\frac{75}{100}}\\ =\frac{6}{75}\\ =\frac{2}{25}\end{array}$

Thus, the probability that bill is rejected provided that bill is legal is $\frac{2}{25}$ .

c.

To determine

To calculate: The probability that a counterfeit bill is not rejected.

Answer to Problem 13CFU

The probability that a counterfeit bill is not rejected is $\frac{1}{25}$ .

Explanation of Solution

Given information:

100 $1- bills were tested and 25 of the bills were counterfeit.

$\begin{array}{ccc}\text{Bill}& \text{Accepted}& \text{Rejected}\\ \text{Legal}& 69& 6\\ \text{Counterfeit }& 1& 24\end{array}$

Formula used:

Probability of an event E is $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Given an event Q, the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

Calculation:

Consider the provided information that 100 $1- bills were tested and 25 of the bills were counterfeit.

$\begin{array}{ccc}\text{Bill}& \text{Accepted}& \text{Rejected}\\ \text{Legal}& 69& 6\\ \text{Counterfeit }& 1& 24\end{array}$

Total number of bills are 100.

Let A be an event that bill is not rejected and B be an event that bill is counterfeit.

Recall that probability of an event E is $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Probability that bill is counterfeit,

  $\begin{array}{c}P\left(B\right)=\frac{24+1}{100}\\ =\frac{25}{100}\end{array}$

Probability that bill is not rejected and bill is counterfeit is,

  $P\left(A\text{ and B}\right)=\frac{1}{100}$

Recall that given an event Q, the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

The probability that a counterfeit bill is not rejected.

  $\begin{array}{c}P\left(A|B\right)=\frac{\frac{1}{100}}{\frac{25}{100}}\\ =\frac{1}{25}\end{array}$

Thus, the probability that a counterfeit bill is not rejected is $\frac{1}{25}$ .