Chapter 4, Problem 4.3.16RE

Factory Output A manufacturing company has three factories: X, Y, and Z. The daily output of each is shown here.

If 1 item is selected at random, find these probabilities.

a. It was manufactured at factory X or is a stereo.

b. It was manufactured at factory Y or factory Z.

c. It is a TV or was manufactured at factory Z.

To determine

The probability of selecting an item produced by factory X or it is a stereo.

Answer to Problem 4.3.16RE

The probability of selecting an item produced by factory X or it is a stereo is 0.548.

Explanation of Solution

Given info:

A company has three manufacturing units X, Y and Z. The three units produce televisions and stereos.

Calculation:

Let A be the event of selecting an item manufactured by unit X.

The probability of any event A is given below:

$\begin{array}{c}P\left(A\right)=\frac{\text{Number of outcomes in }A}{\text{Total number of outcomes in the sample space}}\\ =\frac{n\left(A\right)}{n\left(S\right)}\end{array}$

$\begin{array}{c}P\left(\begin{array}{l}\text{Selecting an item}\\ \text{manufactured by X}\end{array}\right)=\frac{18+6}{18+6+32+20+15+13}\\ =\frac{24}{104}\\ =0.231\end{array}$

Thus, the probability of selecting an item manufactured by unit X is 0.231.

Let B be the event of selecting a stereo.

The probability of any event B is given below:

$\begin{array}{c}P\left(\text{Selecting a stereo}\right)=\frac{6+20+13}{18+32+15+6+20+13}\\ =\frac{39}{104}\\ =0.375\end{array}$

Thus, the probability of selecting a stereo is 0.375.

Addition rule for dependent events:

If two events A and B are said to be dependent, then the addition of events A and B is,

$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$

The number of items common between unit X and stereo is 6.

$\begin{array}{c}P\left(A\cap B\right)=\frac{6}{104}\\ =0.058\end{array}$

The probability of selecting an item produced by unit X or it is a stereo is given below,

$\begin{array}{c}P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)\\ =0.231+0.375-0.058\\ =0.548\end{array}$

Thus, the probability of selecting an item produced by unit X or it is a stereo is 0.548.

To determine

The probability of selecting an item produced by unit Y or by unit Z.

Answer to Problem 4.3.16RE

The probability of selecting anitem produced by unit X or by unit Y is 0.77.

Explanation of Solution

Calculation:

Let C be the event of selecting an item manufactured by unit Y.

The probability of any event C is given below:

$\begin{array}{c}P\left(C\right)=\frac{\text{Number of outcomes in }C}{\text{Total number of outcomes in the sample space}}\\ =\frac{n\left(C\right)}{n\left(S\right)}\end{array}$

The probability of selecting an item manufactured by unit Y is given below:

$\begin{array}{c}P\left(\begin{array}{l}\text{Selecting an item}\\ \text{manufactured by Y}\end{array}\right)=\frac{32+20}{104}\\ =\frac{52}{104}\\ =0.5\end{array}$

Thus, the probability of selecting an item manufactured by unit Y is 0.5.

Let D be the event of selecting an item manufactured by unit Z.

$\begin{array}{c}P\left(\begin{array}{l}\text{Selecting an item}\\ \text{manufactured by Z}\end{array}\right)=\frac{15+13}{104}\\ =\frac{28}{104}\\ =0.27\end{array}$

Thus, the probability of selecting an item manufactured by unit Z is 0.27.

Addition rule for independent events:

If two events A and B are said to be independent, then the addition of events A and B is,

$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)$

The probability of selecting an item produced by unit Y or by unit Z is given below,

$\begin{array}{c}P\left(C\cup D\right)=P\left(C\right)+P\left(D\right)\\ =0.5+0.27\\ =0.77\end{array}$

Thus, the probability of selecting an item produced by unit Y or by unit Z is 0.77.

To determine

The probability of selecting a TV or a unit produced by unit Z.

Answer to Problem 4.3.16RE

The probability of selecting a TV or a unit produced by unit Z is0.769.

Explanation of Solution

Calculation:

Let E denote the event of selecting a TV.

The probability of any event E is given below:

$\begin{array}{c}P\left(\text{Selecting a TV}\right)=\frac{18+32+15}{18+32+15+6+20+13}\\ =\frac{65}{104}\\ =0.625\end{array}$

Thus, the probability of selecting a TV is 0.625.

The number of items common between TV and unit Z is 15.

$\begin{array}{c}P\left(D\cap E\right)=\frac{15}{104}\\ =0.144\end{array}$

The probability of selecting a TV or a unit produced by unit Z is given below:

$\begin{array}{c}P\left(D\cup E\right)=P\left(D\right)+P\left(E\right)\\ =0.625+0.144\\ =0.769\end{array}$

Thus, the probability of selecting a TV or a unit produced by unit Z is 0.769.