Chapter 2.2, Problem 40PPS

(a)

To determine

To Find: The Venn diagram to show the results of the survey.

Answer to Problem 40PPS

The required Venn diagram is shown in Figure 2

Explanation of Solution

Given:

The travel agency survey 70 of the clients that visited Europe about the international travel of the 70 clients those who visited Europe, 60 had travelled to the England, France or both. Of those 60 clients, 45 visited England and 50 visited to the France.

Calculation:

Consider the Venn diagram shown in Figure 1

  

Figure 1

Consider the required system of equations is,

  $\begin{array}{l}x+y+z=60\\ x+y=45\\ y+z=50\end{array}$

Then, add the first and the second equation and subtract the third equation from it this gives,

  $y=35$

Substitute $y=35$ in the second equation as,

  $\begin{array}{l}x+35=45\\ x=10\end{array}$

Substitute $y=35$ in the third equation as,

  $\begin{array}{l}35+z=50\\ z=15\end{array}$

The required Venn diagram is shown in Figure 2

  

Figure 2

(b)

To determine

To Find: Thecompound statement to show the area of the Venn diagram that include the compound statement on the Venn diagram.

Answer to Problem 40PPS

The compound statement to show the area of the Venn diagram that include the compound statement on the Venn diagram is explained and the required Venn diagram is shown in Figure 3.

Explanation of Solution

Consider that the value of shows the number of client who visited the England and the client who visited France is represented by $q$ . The client those who does not visited any of the countries represented by the $~p\vee ~q$ and the client who visited both he country is represented by the $p\wedge q$ . The clients who visited England excludes the client those who visited France excludes the clients who went to the France or $~q$ . The clients with only the England visits is represented by $p\wedge ~q$ . The clients who visited the France only have to be excluded the clients that went to the England or $~q$ . Thus, the France only clients are represented by the $p\wedge ~q$ .

The required Venn diagram is shown in Figure 3

  

Figure 3

(c)

To determine

To Find: The probability that a random chosen participant in the survey will have the visited both the England and France.

Answer to Problem 40PPS

The required probability is $\frac{1}{2}$ .

Explanation of Solution

Since 35 of those surveyed have visited the England and France both and there are total of 70 survey participants, thus the probability that the randomly chosen client has visited both England and France is,

  $\begin{array}{c}P=\frac{35}{70}\\ =\frac{1}{2}\end{array}$