Question

A middle school counselor, attempting to correlate school performance with leisure interests, found that of a group of students, 36 hadseen Movie A, 31 had seen Movie B, 28 had seen Movie C, 17 had seen Movies A and B. 12 had seen Movies A and C, 11 had seenMovies B and C, 4 had seen all three films, and 5 had seen none of the three films. Use a Venn diagram to complete parts (a) through(c) belowComplete the Venn diagram. Let A be the set of students that saw Movie A, B be the set of students that saw Movie B, and C be theset of students that saw Movie C.DCoi(a) How many students had seen Movie C only?student(s)(Simplify your answer.)(b) How many students had seen exactly two of the films?student(s)(Simplify your answer.)(c) How many students were surveyed?student(s)Enter your answer in each of the answer boxes.

Answer 1

a) 9 students have seen Movie C only.

b) 28 students have seen exactly two of the films.

c) 64 students in total wre surveyed.

Explanation

First thing to clarify is that for questions on sets, when they give the number of students that have seen movie A for example, this consists of those that have seen movie A only, those that have seen movie A and movie B, those that have seen movie A and movie C and those that have seen movie A, movie B and movie C. In mathematical terms, this means

n(A) = n(A n B' n C') + n(A n B n C') + n(A n B' n C) + n(A n B n C)

Also not that when they give the number of people who have seen movie A and movie B, this number consists of those that have seen movie A and movie B only and those that have seen movies A, B and C. In mathematical terms, this means

n(A n B) = n(A n B n C') + n(A n B n C)

So, after noting this, I'll start the interpretations.

36 have seen Movie A.

n(A) = 36

31 had seen Movie B.

n(B) = 31

28 had seen Movie C,

n(C) = 28

17 had seen Movies A and B.

n(A n B) = 17

12 had seen Movies A and C.

n(A n C) = 12

11 had seen Movies B and C.

n(B n C) = 11

4 had seen all three films.

n(A n B n C) = 4

5 had seen none of the three films.

n(A u B u C)' = n(A' n B' n C') = 5

So, to start with, we first find those that have seen ONLY two of the movies,

- Those that have seen movie A and movie B ONLY = n(A n B n C')

n(A n B) = n(A n B n C') + n(A n B n C)

17 = n(A n B n C') - 4

n(A n B n C') = 17 - 4 = 13

- Those that have seen movie A and movie C ONLY = n(A n B' n C)

n(A n C) = n(A n B' n C) + n(A n B n C)

12 = n(A n B' n C) - 4

n(A n B' n C) = 12 - 4 = 8

- Those that have seen movie B and movie C ONLY = n(A' n B n C)

n(B n C) = n(A' n B n C) + n(A n B n C)

11 = n(A' n B n C) - 4

n(A' n B n C) = 11 - 4 = 7

Then, we can then calculate those that have watched ONLY one movie

- Those that have seen only movie A = n(A n B' n C')

n(A) = n(A n B' n C') + n(A n B n C') + n(A n B' n C) + n(A n B n C)

36 = n(A n B' n C') + 13 + 8 + 4

n(A n B' n C') = 36 - 13 - 8 - 4 = 11

- Those that have seen only movie B = n(A' n B n C')

n(B) = n(A' n B n C') + n(A n B n C') + n(A' n B n C) + n(A n B n C)

31 = n(A' n B n C') + 13 + 7 + 4

n(A' n B n C') = 31 - 13 - 7 - 4 = 7

- Those that have seen only movie C = n(A' n B' n C)

n(C) = n(A' n B' n C) + n(A n B' n C) + n(A' n B n C) + n(A n B n C)

28 = n(A n B' n C') + 8 + 7 + 4

n(A' n B' n C) = 28 - 8 - 7 - 4 = 9

a) How many students have seen Movie C only?

From the Venn Diagram and the calculations, only 9 students have seen Movie C only.

b) How many students have seen exactly two of the films?

From the Venn Diagram and the calculations, this will be the sum of students that have seen Movies A and B ONLY, Movies A and C ONLY and Movies B and C ONLY.

= 13 + 7 + 8

= 28 students

c) How many students were surveyed in total?

From the Venn Diagram, we can just add all the numbers together since we have seperated them into 'ONLY' numbers.

Total number of students surveyed = 11 + 7 + 9 + 13 + 8 + 7 + 4 + 5

= 64

Hope this Helps!!!