In a survey of 100 people it was found that 65 read magazine [tex]$A$[/tex], 45 read magazine [tex]$B$[/tex], 40 read magazine [tex]$C$[/tex], 25 read magazine [tex]$A$[/tex] as well as [tex]$B$[/tex]; 20 read magazine [tex]$A$[/tex] as well as [tex]$C$[/tex], 15 read magazine [tex]$B$[/tex] as well as [tex]$C$[/tex]; 5 read all three magazines and 5 don't read any of them.
i) Find the value of [tex]$n(A \cap B \cap C)$[/tex]
ii) Show the information in a Venn diagram.
iii) Find the value of [tex]$n(A \cup B \cup C)$[/tex]
iv) In every 20 people, one does not read all three magazines. Justify it.
Answer (2)
The number of people who read all three magazines is n ( A c a pB c a pC ) = 5 .
Using the principle of inclusion-exclusion, the number of people who read at least one magazine is n ( A c u pB c u pC ) = 95 .
The Venn diagram is constructed with the calculated values for each region.
The statement that in every 20 people, one does not read all three magazines is justified.
5 , 95
Explanation
Analyze the problem and data Let's break down this survey problem step by step. We're given information about how many people read magazines A, B, and C, along with overlaps between them, and we need to find some specific values and represent the data in a Venn diagram.
Find the value of n ( A c a pB c a pC ) i) The question asks for the value of n ( A c a pB c a pC ) , which represents the number of people who read all three magazines. The problem statement directly tells us that 5 people read all three magazines. Therefore, n ( A c a pB c a pC ) = 5 .
Show the information in a Venn diagram ii) To create the Venn diagram, we start with the intersection of all three sets, which is 5. Next, we calculate the number of people who read exactly two magazines.
n ( A c a pB ) = 25 , so the number of people who read only A and B is 25 − 5 = 20 .
n ( A c a pC ) = 20 , so the number of people who read only A and C is 20 − 5 = 15 .
n ( B c a pC ) = 15 , so the number of people who read only B and C is 15 − 5 = 10 .
Now, we calculate the number of people who read only one magazine:
n ( A ) = 65 , so the number of people who read only A is 65 − 20 − 15 − 5 = 25 .
n ( B ) = 45 , so the number of people who read only B is 45 − 20 − 10 − 5 = 10 .
n ( C ) = 40 , so the number of people who read only C is 40 − 15 − 10 − 5 = 10 .
Finally, 5 people read none of the magazines.
Find the value of n ( A c u pB c u pC ) iii) To find n ( A c u pB c u pC ) , which is the number of people who read at least one of the magazines, we can use the principle of inclusion-exclusion:
n ( A c u pB c u pC ) = n ( A ) + n ( B ) + n ( C ) − n ( A c a pB ) − n ( A c a pC ) − n ( B c a pC ) + n ( A c a pB c a pC )
Plugging in the values, we get:
n ( A c u pB c u pC ) = 65 + 45 + 40 − 25 − 20 − 15 + 5 = 95
Alternatively, since 5 people read none of the magazines, the number of people who read at least one magazine is 100 − 5 = 95 .
Justify the statement iv) To justify the statement that in every 20 people, one does not read all three magazines, we need to find the number of people who do not read all three magazines. Since 5 people read all three magazines, 100 − 5 = 95 people do not read all three magazines. Now, we divide the total number of people (100) by 20, which gives us 5 groups of 20 people each. In each group of 20, we need to check if at least one person does not read all three magazines. Since 95 people do not read all three magazines, and there are 5 groups of 20, on average, there are 95/5 = 19 people in each group who do not read all three magazines. Since 19 is greater than 1, the statement is justified.
State the final answer Therefore, the final answers are: i) n ( A c a pB c a pC ) = 5 ii) The Venn diagram is constructed as described above. iii) n ( A c u pB c u pC ) = 95 iv) The statement is justified.
Examples
Venn diagrams and set theory are useful in market research to understand customer preferences. For example, a company might survey customers about their preferences for different product features (A, B, C). By analyzing the overlaps (intersections) and unions of these preferences, the company can identify target customer segments and tailor their marketing strategies accordingly. If a survey reveals that 65% of customers prefer feature A, 45% prefer feature B, and 40% prefer feature C, and there are overlaps, the company can use the inclusion-exclusion principle to determine the total percentage of customers who prefer at least one of the features. This helps in making informed decisions about product development and marketing campaigns.
The number of people who read all three magazines is 5, and the total number of people who read at least one magazine is 95. The statement about every 20 people is justified as on average there are 19 people in each group who do not read all three magazines. A Venn diagram can illustrate these results visually.
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