Chapter 2.1, Problem 4E

Three times each day, a quality engineer samples a component from a recently manufactured batch and tests it. Each part is classified as conforming (suitable for its intended use), downgraded (unsuitable for the intended purpose but usable for another purpose), or scrap (not usable). An experiment consists of recording the categories of the three parts tested in a particular day.

1. a. List the 27 outcomes in the sample space.
2. b. Let A be the event that all the parts fall into the same category. List the outcomes in A.
3. c. Let B be the event that there is one part in each category. List the outcomes in B.
4. d. Let C be the event that at least two parts are conforming. List the outcomes in C.
5. e. List the outcomes in A Ç C.
6. f. List the outcomes in A È B.
7. g. List the outcomes in A Ç C^c.
8. h. List the outcomes in A^c Ç C.
9. i. Are events A and C mutually exclusive? Explain.
10. j. Are events B and C mutually exclusive? Explain.

To determine

List all 27 outcomes in the sample space.

Answer to Problem 4E

The all 27 outcomes in the sample space is,

$\left\{\begin{array}{l}CCC,CCD,CCS,CDC,CDD,CDS,CSC,CSD,CSS,\\ DCC,DCD,DCS,DDC,DDD,DDS,DSC,DSD,DSS,\\ SCC,SCD,SCS,SDC,SDD,SDS,SSC,SSD,SSS\end{array}\right\}$.

Explanation of Solution

Given info:

Three times each day, a quality engineer samples a component from a recently manufactured batch and tests it. Each part is classified as conforming (suitable for its intended use), downgraded (unsuitable for the intended purpose but usable for another purpose), or scrap (not usable). An experiment consists of recording the categories of the three parts tested in a particular day.

Calculation:

Sample space:

The set of all possible outcomes of an experiment is called the sample space.

Let C represents conforming, D represents downgraded and S represents scrap.

Here, it is observed that the number of parts tested in a particular day is 3 and number of times tested in each day is 3.

Threfore, the number of all possible outcomes of the experiment is 28 $\left(=3^3\right)$.

Thus, the set of all outcomes of this experiment is,

$\left\{\begin{array}{l}CCC,CCD,CCS,CDC,CDD,CDS,CSC,CSD,CSS,\\ DCC,DCD,DCS,DDC,DDD,DDS,DSC,DSD,DSS,\\ SCC,SCD,SCS,SDC,SDD,SDS,SSC,SSD,SSS\end{array}\right\}$

To determine

List the outcomes in A if A is the event that all the parts fall into the same category.

Answer to Problem 4E

The outcomes in A are $\left\{CCC,DDD,SSS\right\}$.

Explanation of Solution

Calculation:

Here, A is the event that all the parts fall into the same category.

The possible outcomes for all the parts fall into the same category are,

$\left\{CCC,DDD,SSS\right\}$.

Thus, the outcomes in A are $\left\{CCC,DDD,SSS\right\}$.

To determine

List the outcomes in B if B is the event that there is one part in each category.

Answer to Problem 4E

The outcomes in B are $\left\{CDS,CSD,DCS,DSC,SCD,SDC\right\}$.

Explanation of Solution

Calculation:

Here, B is the event that there is one part in each category.

The possible outcomes for there is one part in each category are,

$\left\{CDS,CSD,DCS,DSC,SCD,SDC\right\}$.

Thus, the outcomes in B are $\left\{CDS,CSD,DCS,DSC,SCD,SDC\right\}$.

To determine

List the outcomes in C if C is the event that at least two parts are confirming.

Answer to Problem 4E

The outcomes in C are $\left\{CCD,CCS,CDC,CSC,DCC,SCC,CCC\right\}$.

Explanation of Solution

Calculation:

Here, C is the event that at least two parts are confirming.

The possible outcomes for at least two parts are confirming are,

$\left\{CCD,CCS,CDC,CSC,DCC,SCC,CCC\right\}$.

Thus, the outcomes in C are $\left\{CCD,CCS,CDC,CSC,DCC,SCC,CCC\right\}$.

To determine

List the outcomes in $A\cap C$.

Answer to Problem 4E

The outcomes in $A\cap C$ are $\left\{CCC\right\}$.

Explanation of Solution

Calculation:

Intersection:

The intersection of two events A and B is the set of outcoems that belong both to A and to B. It is denoted by $A\cap B$.

From part (b) ann (d),

$A=\left\{CCC,DDD,SSS\right\}$

$C=\left\{CCD,CCS,CDC,CSC,DCC,SCC,CCC\right\}$

Here, $A\cap C$ represents is the set of outcoems that belong both to A and to C.

The possible outcomes for $A\cap C$ is,

$\left\{CCC\right\}$.

Thus, the outcomes in $A\cap C$ are $\left\{CCC\right\}$.

To determine

List the outcomes in $A\cup B$.

Answer to Problem 4E

The outcomes in $A\cup B$ are $\left\{CCC,DDD,SSS,CDS,CSD,DCS,DSC,SCD,SDC\right\}$.

Explanation of Solution

Calculation:

Union:

The unions of two events A and B is the set of outcoems that belong either to A, to B, or both. It is denoted by $A\cup B$.

From part (b) ann (d),

$A=\left\{CCC,DDD,SSS\right\}$

$B=\left\{CDS,CSD,DCS,DSC,SCD,SDC\right\}$

Here, $A\cup B$ represents is the set of outcoems that belong either to A, to B, or both.

The possible outcomes for $A\cup B$ is,

$\left\{CCC,DDD,SSS,CDS,CSD,DCS,DSC,SCD,SDC\right\}$.

Thus, the outcomes in $A\cap C$ are $\left\{CCC,DDD,SSS,CDS,CSD,DCS,DSC,SCD,SDC\right\}$.

To determine

List the outcomes in $A\cap C^c$.

Answer to Problem 4E

The outcomes in $A\cap C^c$ are $\left\{DDD,SSS\right\}$.

Explanation of Solution

Calculation:

Complement:

The complement of an event A is the set of outcomes that do not belong to A. It is denoted by $A^c$.

Here, $C^c$ represents is the set of outcomes that do not belong to C and $A\cap C^c$ represents the set of outcoems that belong both to A and to $C^c$.

$C^c=\left\{\begin{array}{l}CDD,CDS,CSC,CSD,CSS,DCD,DCS,\\ DDC,DDD,DDS,DSC,DSD,DSS,SCD,\\ SCS,SDC,SDD,SDS,SSC,SSD,SSS\end{array}\right\}$

The possible outcomes for $A\cap C^c$ is,

$\left\{DDD,SSS\right\}$

Thus, the outcomes in $A\cap C^c$ are $\left\{DDD,SSS\right\}$.

To determine

List the outcomes in $A^c\cap C$.

Answer to Problem 4E

The outcomes in $A^c\cap C$ are $\left\{CCD,CCS,CDC,CSC,DCC,SCC\right\}$.

Explanation of Solution

Calculation:

Complement:

The complement of an event A is the set of outcomes that do not belong to A. It is denoted by $A^c$.

Here, $A^c$ represents is the set of outcomes that do not belong to A and $A^c\cap C$ represents the set of outcoems that belong both to $A^c$ and to C.

$A^c=\left\{\begin{array}{l}CCD,CCS,CDC,CDD,CDS,CSC,CSD,CSS,\\ DCC,DCD,DCS,DDC,DDS,DSC,DSD,DSS,\\ SCC,SCD,SCS,SDC,SDD,SDS,SSC,SSD\end{array}\right\}$

The possible outcomes for $A^c\cap C$ is,

$\left\{CCD,CCS,CDC,CSC,DCC,SCC\right\}$

Thus, the outcomes in $A\cap C^c$ are $\left\{CCD,CCS,CDC,CSC,DCC,SCC\right\}$.

To determine

Check whether the events A and C are mutually exclusive.

Answer to Problem 4E

No, the events A and C are not mutually exclusive.

Explanation of Solution

Calculation:

Mutually exclusive:

The events A and B are mutually exclusive if they have no common outcomes.

From part (b) and (d),

$A=\left\{CCC,DDD,SSS\right\}$

$C=\left\{CCD,CCS,CDC,CSC,DCC,SCC,CCC\right\}$

Here, it is observed that the outcome $\left\{CCC\right\}$ is common in both outcomes A and C.

Therefore, the events A and C are not mutually exclusive.

To determine

Check whether the events B and C are mutually exclusive.

Answer to Problem 4E

Yes, the events B and C are mutually exclusive.

Explanation of Solution

Calculation:

From part (c) and (d),

$B=\left\{CDS,CSD,DCS,DSC,SCD,SDC\right\}$

$C=\left\{CCD,CCS,CDC,CSC,DCC,SCC,CCC\right\}$

Here, it is observed that the no outcome is common in both outcomes B and C.

Therefore, the events B and C are mutually exclusive.