Chapter 4.6, Problem 4.40E

An experiment can result in one of five equally likely simple events, $E_1,E_2,\mathrm{...},E_5$ . Events A, B, and C are defined as follows:

$\begin{array}{l}A:E_1,E_{3}P\left(A\right)=.4\\ B:E_1,E_2,E_4,E_{5}P\left(B\right)=.8\\ C:E_3,E_{4}P\left(C\right)=.4\end{array}$

Find the probabilities associated with the following events by listing the simple events in each.

a. $A^c$   b. $A\cap B$ <  c. $B\cap C$

d. $A\cup B$   e. $B|C$   f. $A|B$

g. $A\cup B\cup C$   h. ${\left(A\cap B\right)}^c$

To determine

To find:the probability associated with the given events $A^c$ by listing the simple events

Answer to Problem 4.40E

  $P\left(A^c\right)=0.6$

Explanation of Solution

Given:

An experiment can result in one of the five equally likely simple events, $E_1,E_2,E_3,E_4,E_5$

Events A, B and C are defined as follows:

  $\begin{array}{l}A:E_1,E_{3}P\left(A\right)=.4\\ B:E_1,E_2,E_4,E_{5}P\left(B\right)=.8\\ C:E_3,E_{4}P\left(C\right)=.4\end{array}$

Calculation:

Since, all simple events are equally likely,

  $P\left(E_1\right)=0.2$ , ${\forall }_1=\left\{1,2,3,4,5\right\}$

  $\begin{array}{l}{A}^c=\left\{E_2,E_4,E_5\right\}\\ P\left(A^c\right)=P\left(E_2\right)+P\left(E_4\right)+P\left(E_5\right)\\ P\left(A^c\right)=0.6\end{array}$

Hence,the probability associated with the given events is $P\left(A^c\right)=0.6$ .

To determine

To find: the probability associated with the given events $A\cap B$ by listing the simple events

Answer to Problem 4.40E

  $P\left(A\cap B\right)=0.2$

Explanation of Solution

Given:

An experiment can result in one of the five equally likely simple events, $E_1,E_2,E_3,E_4,E_5$

Events A, B and C are defined as follows:

  $\begin{array}{l}A:E_1,E_{3}P\left(A\right)=.4\\ B:E_1,E_2,E_4,E_{5}P\left(B\right)=.8\\ C:E_3,E_{4}P\left(C\right)=.4\end{array}$

Calculation:

Since, all simple events are equally likely,

  $P\left(E_1\right)=0.2$ , ${\forall }_1=\left\{1,2,3,4,5\right\}$

  $\begin{array}{l}A\cap B=\left\{E_1\right\}\\ P\left(A\cap B\right)=P\left(E_1\right)\\ P\left(A\cap B\right)=0.2\end{array}$

Hence,the probability associated with the given events is $P\left(A\cap B\right)=0.2$ .

To determine

To find: the probability associated with the given events $B\cap C$ by listing the simple events

Answer to Problem 4.40E

  $P\left(B\cap C\right)=0.2$

Explanation of Solution

Given:

An experiment can result in one of the five equally likely simple events, $E_1,E_2,E_3,E_4,E_5$

Events A, B and C are defined as follows:

  $\begin{array}{l}A:E_1,E_{3}P\left(A\right)=.4\\ B:E_1,E_2,E_4,E_{5}P\left(B\right)=.8\\ C:E_3,E_{4}P\left(C\right)=.4\end{array}$

Calculation:

Since, all simple events are equally likely,

  $P\left(E_1\right)=0.2$ , ${\forall }_1=\left\{1,2,3,4,5\right\}$

  $\begin{array}{l}B\cap C=\left\{E_4\right\}\\ P\left(B\cap C\right)=P\left(E_4\right)\\ P\left(B\cap C\right)=0.2\end{array}$

Hence,the probability associated with the given events is $P\left(B\cap C\right)=0.2$ .

To determine

To find: the probability associated with the given events $A\cup B$ by listing the simple events

Answer to Problem 4.40E

  $P\left(A\cup B\right)=1$

Explanation of Solution

Given:

An experiment can result in one of the five equally likely simple events, $E_1,E_2,E_3,E_4,E_5$

Events A, B and C are defined as follows:

  $\begin{array}{l}A:E_1,E_{3}P\left(A\right)=.4\\ B:E_1,E_2,E_4,E_{5}P\left(B\right)=.8\\ C:E_3,E_{4}P\left(C\right)=.4\end{array}$

Calculation:

Since, all simple events are equally likely,

  $P\left(E_1\right)=0.2$ , ${\forall }_1=\left\{1,2,3,4,5\right\}$

  $\begin{array}{l}A\cup B=\left\{E_1,E_2,E_3,E_4,E_5\right\}\\ P\left(A\cup B\right)=P\left(E_1\right)+P\left(E_2\right)+P\left(E_3\right)+P\left(E_4\right)+P\left(E_5\right)\\ P\left(A\cup B\right)=0.2+0.2+0.2+0.2+0.2\\ P\left(A\cup B\right)=1\end{array}$

Hence,the probability associated with the given events is $P\left(A\cup B\right)=1$ .

To determine

To find: the probability associated with the given events $B|C$ by listing the simple events

Answer to Problem 4.40E

  $P\left(B|C\right)=0.5$

Explanation of Solution

Given:

An experiment can result in one of the five equally likely simple events, $E_1,E_2,E_3,E_4,E_5$

Events A, B and C are defined as follows:

  $\begin{array}{l}A:E_1,E_{3}P\left(A\right)=.4\\ B:E_1,E_2,E_4,E_{5}P\left(B\right)=.8\\ C:E_3,E_{4}P\left(C\right)=.4\end{array}$

Calculation:

Simply events in $B|C$ is $\left\{E_4\right\}$

  $\begin{array}{l}P\left(B|C\right)=\frac{P\left(E_4\right)}{P\left(C\right)}\\ P\left(B|C\right)=\frac{0.2}{0.4}\\ P\left(B|C\right)=0.5\end{array}$

Hence,the probability associated with the given events is $P\left(B|C\right)=0.5$ .

To determine

To find: the probability associated with the given events $A|B$ by listing the simple events

Answer to Problem 4.40E

  $P\left(A|B\right)=0.25$

Explanation of Solution

Given:

An experiment can result in one of the five equally likely simple events, $E_1,E_2,E_3,E_4,E_5$

Events A, B and C are defined as follows:

  $\begin{array}{l}A:E_1,E_{3}P\left(A\right)=.4\\ B:E_1,E_2,E_4,E_{5}P\left(B\right)=.8\\ C:E_3,E_{4}P\left(C\right)=.4\end{array}$

Calculation:

Simply events in $A|B$ is $\left\{E_3\right\}$

  $\begin{array}{l}P\left(A|B\right)=\frac{P\left(E_3\right)}{P\left(B\right)}\\ P\left(A|B\right)=\frac{0.2}{0.8}\\ P\left(A|B\right)=0.25\end{array}$

Hence,the probability associated with the given events is $P\left(A|B\right)=0.25$ .

To determine

To find: the probability associated with the given events $A\cup B\cup C$ by listing the simple events

Answer to Problem 4.40E

  $P\left(A\cup B\cup C\right)=1$

Explanation of Solution

Given:

An experiment can result in one of the five equally likely simple events, $E_1,E_2,E_3,E_4,E_5$

Events A, B and C are defined as follows:

  $\begin{array}{l}A:E_1,E_{3}P\left(A\right)=.4\\ B:E_1,E_2,E_4,E_{5}P\left(B\right)=.8\\ C:E_3,E_{4}P\left(C\right)=.4\end{array}$

Calculation:

Since, all simple events are equally likely,

  $P\left(E_1\right)=0.2$ , ${\forall }_1=\left\{1,2,3,4,5\right\}$

  $\begin{array}{l}A\cup B\cup C=\left\{E_1,E_2,E_3,E_4,E_5\right\}\\ P\left(A\cup B\cup C\right)=P\left(E_1\right)+P\left(E_2\right)+P\left(E_3\right)+P\left(E_4\right)+P\left(E_5\right)\\ P\left(A\cup B\cup C\right)=0.2+0.2+0.2+0.2+0.2\\ P\left(A\cup B\cup C\right)=1\end{array}$

Hence,the probability associated with the given events is $P\left(A\cup B\cup C\right)=1$ .

To determine

To find: the probability associated with the given events ${\left(A\cap B\right)}^c$ by listing the simple events

Answer to Problem 4.40E

  $P\left(A\cap B\right)=0.2$

Explanation of Solution

Given:

An experiment can result in one of the five equally likely simple events, $E_1,E_2,E_3,E_4,E_5$

Events A, B and C are defined as follows:

  $\begin{array}{l}A:E_1,E_{3}P\left(A\right)=.4\\ B:E_1,E_2,E_4,E_{5}P\left(B\right)=.8\\ C:E_3,E_{4}P\left(C\right)=.4\end{array}$

Calculation:

Since, all simple events are equally likely,

  $P\left(E_1\right)=0.2$ , ${\forall }_1=\left\{1,2,3,4,5\right\}$

  $\begin{array}{l}{\left(A\cap B\right)}^c=\left\{E_1\right\}\\ P{\left(A\cap B\right)}^c=P\left(E_1,E_3,E_4,E_5\right)\\ P{\left(A\cap B\right)}^c=P\left(E_1\right)+P\left(E_3\right)+P\left(E_4\right)+P\left(E_5\right)\\ P{\left(A\cap B\right)}^c=0.2+0.2+0.2+0.2\\ P{\left(A\cap B\right)}^c=0.8\end{array}$

Hence,the probability associated with the given events is $P{\left(A\cap B\right)}^c=0.8$ .