Chapter 4, Problem 4.102SE

a.

To determine

To list: the simple events in $S$ .

Answer to Problem 4.102SE

  $S=\left\{\begin{array}{l}{A}_1{A}_2,A_1{C}_1,A_1{C}_2,A_2{A}_1,A_2{C}_1,A_2{C}_1\\ C_1{A}_1,C_1{A}_2,C_1{C}_2,C_2{A}_1,C_2{A}_2,C_2{C}_1\end{array}\right\}$

Explanation of Solution

Given:

The given statement is:

Two cold tablets are unintentionally placed in a box containing two aspirin tablets.

The four tablets are identical in appearance in which one tablet is selected at random from the box.

Calculation:

According to the question,

Let $C_1\text{ and }{C}_2$ be two cold tablets and $A_1\text{ and }{A}_2$ be the two aspirin tablets.

Let $xy$ represent the event that man $x\text{ and }y$ are swallowed by first and second patient, respectively.

Therefore, the simple events in sample space is:

  $S=\left\{\begin{array}{l}{A}_1{A}_2,A_1{C}_1,A_1{C}_2,A_2{A}_1,A_2{C}_1,A_2{C}_1\\ C_1{A}_1,C_1{A}_2,C_1{C}_2,C_2{A}_1,C_2{A}_2,C_2{C}_1\end{array}\right\}$

b.

To determine

To find:the event $A$ that the first patient obtained a cold tablet.

Answer to Problem 4.102SE

  $A=\left\{C_1{A}_1,C_1{A}_2,C_1,C_2,C_2{A}_1,C_2{A}_2,C_2{C}_1\right\}$

Explanation of Solution

Given:

The given statement is:

Two cold tablets are unintentionally placed in a box containing two aspirin tablets.

The four tablets are identical in appearance in which one tablet is selected at random from the box.

Calculation:

According to the question,

Let $C_1\text{ and }{C}_2$ be two cold tablets and $A_1\text{ and }{A}_2$ be the two aspirin tablets.

Let $xy$ represent the event that man $x\text{ and }y$ are swallowed by first and second patient, respectively.

As the sample space are from part a is $S=\left\{\begin{array}{l}{A}_1{A}_2,A_1{C}_1,A_1{C}_2,A_2{A}_1,A_2{C}_1,A_2{C}_1\\ C_1{A}_1,C_1{A}_2,C_1{C}_2,C_2{A}_1,C_2{A}_2,C_2{C}_1\end{array}\right\}$

Therefore, theevent $A$ that the first patient obtained a cold tablet where $x\in \left\{C_1,C_2\right\}$ .

Hence, the event $A$ that the first patient obtained a cold tablet is $A=\left\{C_1{A}_1,C_1{A}_2,C_1,C_2,C_2{A}_1,C_2{A}_2,C_2{C}_1\right\}$

c.

To determine

To find: the event $B$ that exactly one of the twopatients obtained a cold tablet.

Answer to Problem 4.102SE

  $B=\left\{C_1{A}_1,C_1{A}_2,C_2{A}_1,C_2{A}_2,A_1{C}_1,A_1{C}_2,A_2{C}_1,A_2{C}_2\right\}$

Explanation of Solution

Given:

The given statement is:

Two cold tablets are unintentionally placed in a box containing two aspirin tablets.

The four tablets are identical in appearance in which one tablet is selected at random from the box.

Calculation:

According to the question,

Let $C_1\text{ and }{C}_2$ be two cold tablets and $A_1\text{ and }{A}_2$ be the two aspirin tablets.

Let $xy$ represent the event that man $x\text{ and }y$ are swallowed by first and second patient, respectively.

As the sample space are from part a is $S=\left\{\begin{array}{l}{A}_1{A}_2,A_1{C}_1,A_1{C}_2,A_2{A}_1,A_2{C}_1,A_2{C}_1\\ C_1{A}_1,C_1{A}_2,C_1{C}_2,C_2{A}_1,C_2{A}_2,C_2{C}_1\end{array}\right\}$

Therefore, the event $B$ that the first patient obtained a cold tablet where $x\in \left\{C_1,C_2\right\}$ or $y\in \left\{C_1,C_2\right\}$ , not both.

Hence, the event $B$ that the first patient obtained a cold tablet is $B=\left\{C_1{A}_1,C_1{A}_2,C_2{A}_1,C_2{A}_2,A_1{C}_1,A_1{C}_2,A_2{C}_1,A_2{C}_2\right\}$ .

Hence,

  $P\left(\text{Two men from minority group get least desirable jobs}\right)=\frac{1}{6}$

d.

To determine

To find: the event $C$ that neither patient obtained a cold tablet.

Answer to Problem 4.102SE

  $\left\{A_1{A}_2,A_2{A}_1\right\}$

Explanation of Solution

Given:

The given statement is:

Two cold tablets are unintentionally placed in a box containing two aspirin tablets.

The four tablets are identical in appearance in which one tablet is selected at random from the box.

Calculation:

According to the question,

Let $C_1\text{ and }{C}_2$ be two cold tablets and $A_1\text{ and }{A}_2$ be the two aspirin tablets.

Let $xy$ represent the event that man $x\text{ and }y$ are swallowed by first and second patient, respectively.

As the sample space are from part a is $S=\left\{\begin{array}{l}{A}_1{A}_2,A_1{C}_1,A_1{C}_2,A_2{A}_1,A_2{C}_1,A_2{C}_1\\ C_1{A}_1,C_1{A}_2,C_1{C}_2,C_2{A}_1,C_2{A}_2,C_2{C}_1\end{array}\right\}$

Therefore, the event $C$ that neither patient obtained a cold tablet is $\left\{A_1{A}_2,A_2{A}_1\right\}$

Hence, the event $C$ that neither patient obtained a cold tablet is $\left\{A_1{A}_2,A_2{A}_1\right\}$ .