Question

A travel agency surveyed 70 of their clients who had visited Europe about international travel. Of the 70 clients who had visited Europe, 60 had traveled to England, France, or both. Of those 60 clients, 45 had visited England, and 50 had visited France.

a) Make a Venn diagram to show the results of the survey
b) If p represents a client who has visited England and q represents a client who has visited France, write a compound statement to represent each area of the Venn diagram. Include the compound statements on your Venn diagram.
c) What is the probability that a randomly chosen participant in the survey will have visited both England and France? Explain your reasoning.

Answer 1

The compound statements for each area of the Venn diagram are:

p represents a client who has visited England.

q represents a client who has visited France.

p ∩ q represents the intersection of clients who have visited both England and France.

The probability of a randomly chosen participant having visited both England and France is 0.5 or 50%.

How to make Venn diagram

Let's break this down step by step:

a) Venn Diagram:

First, create a Venn diagram to represent the information given:

Total clients who visited Europe (represented by a rectangle): 70

Clients who visited England (E): 45

Clients who visited France (F): 50

Clients who visited both England and France (E ∩ F): Given as part of the information.

b) Compound Statements:

Using set notation, the compound statements for each area of the Venn diagram are:

p represents a client who has visited England.

q represents a client who has visited France.

p ∩ q represents the intersection of clients who have visited both England and France.

c) Probability of Visiting Both England and France:

The probability of a randomly chosen participant in the survey having visited both England and France can be calculated using the formula for probability:

P(both)= Total number of clients/Number of clients who visited both

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From the Venn diagram, the number of clients who visited both England and France (E ∩ F) is the intersection area. Using the given information, this value is not directly provided but can be calculated using the principle of inclusion-exclusion:

Total clients who visited England or France=Clients who visited England + Clients who visited France − Clients who visited both

60=45+50−Clients who visited both

Clients who visited both=45+50−60=35

P(both)= 35/70 =0.5

Step-by-step explanation:

The probability of a randomly chosen participant having visited both England and France is 0.5 or 50%. This is because out of the 70 participants surveyed, 35 visited both England and France. Therefore, the probability is calculated by dividing the number of clients who visited both countries by the total number of participants.