A local sports bar surveyed 67 of its customers about their favorite teams, and the following information was obtained: 30 people were Alabama fans, 44 people were Texas fans, 23 people were...

A local sports bar surveyed 67 of its customers about their favorite teams, and the following information was obtained: 30 people were Alabama fans, 44 people were Texas fans, 23 people were Michigan fans, 20 people were both Alabama and Michigan fans, 5 people were Alabama and Michigan fans, but not Texas fans, 15 people were Texas and Michigan fans, and 23 were only Texas fans. 

a. How many people were fans of at least one of the teams (Alabama, Michigan, or Texas)?  
b. How many people were fans of all 3 teams?  
c. How many people were fans of none of the teams?  

Asked on by user8042862

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

Use a Venn diagram: Let the rectangle be the universe of people interviewed; U=67.

There are three overlapping circles: label them A for Alabama, T for Texas, and M for Michigan.

From the information given we have the number of people in A=30, in T=44, and in M=23. Also the number of people in the overlap of A and M is 20; 5 of these are not in the intersection of all of the circles, so 15 are in the intersection of all circles.

Since 15 people were Texas and Michigan fans, and the 3-way intersection holds 15, there are no fans of just Texas and Michigan.

Since there are 23 total Michigan fans, 15 in the 3-way intersection and 5 in the intersection with A, 0 in the intersection with T, there must be 3 in M only.

Since there is a total of 44 T fans, 23 of whom are T only, 15 all 3, 0 T and M, that leaves 6 A and T. Since there are 30 total in A, 5 in A and M, 15 in all 3,6 in A and T, that leaves 4 in A only.

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The Venn diagram is filled out as follows:

A only = 4

A and M only = 5

M only = 3

M and T only = 0

T only = 23

A and T only = 6

A,T and M=15

Total within the 3 circles is 56. Total outside the circles is 67-56=11.

(a) There are 56 fans of at least one team.

(b) There are 15 fans of all three.

(c) There are 11 who are not fans of any of the three.

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