Movies A survey of 350 customers was taken at Regal Cinemas in Austin, Texas, regarding the type of movies customers liked. The following information was determined.
• 196 liked dramas.
• 153 liked comedies.
• 88 liked science fiction.
• 59 liked dramas and comedies.
• 37 liked dramas and science fiction.
• 32 liked comedies and science fiction.
• 21 liked all three types of movies
Of the customers surveyed, how many liked
• a) none of these types of movies?
• b) only dramas?
• c) exactly one of these types of movies?
I do not know how to determne how to get the answers the A, B and C nor how to input it into a Venn Diagram
Draw 3 overlapping circles, D, C and SF.
Way in the middle where all three overlap, put 21
Where only D and C overlap, put 59
Where D has no overlap with others pu 196
etc
now you have to work from the inside out
to figure out how many like each 2 or each 1 exclusively
For example those liking only D and C
are
59-21 = 38
D and SF = 37-21 =16
C and SF = 32-21 = 11
then
those liking D alone = 196 - 38-21-16 =
121
those liking C alone = 153-38-21-11 =
83
those liking SF alone = 88-16-11-21 =
40
NOW do the problem
a
none = 350 - 121-38-21-16-83-11-40 = 20
b
121
c
121+83+40 = 244
Check my arithmetic!!!
To find the answers to parts a), b), and c) of the question, we can use a Venn diagram to visualize the information given. A Venn diagram consists of overlapping circles that represent different sets or categories. In this case, we will use three circles to represent dramas, comedies, and science fiction movies.
To construct the Venn diagram, we can start by labeling the regions where the circles overlap based on the given information:
- Let's label the region where all three circles overlap as "Drama, Comedy, and Science Fiction" and write the value of 21 within it.
- Next, let's label the region that overlaps between dramas and comedies (but not science fiction) as "Drama and Comedy" and write the value of 59 within it.
- Similarly, label the region that overlaps between dramas and science fiction (but not comedies) as "Drama and Science Fiction" and write the value of 37 within it.
- Lastly, label the region that overlaps between comedies and science fiction (but not dramas) as "Comedy and Science Fiction" and write the value of 32 within it.
Now, we can determine the number of customers who liked each category:
a) To find the number of customers who liked none of these types of movies, we need to calculate the number of customers outside the three circles. We can start by adding up the values we have written within each region: 21 + 59 + 37 + 32 = 149. Since the total number of customers surveyed is 350, we subtract the sum of the overlapping regions from the total: 350 - 149 = 201. So, 201 customers liked none of the types of movies.
b) To find the number of customers who liked only dramas, we need to subtract the values occurring in regions that include other movie types. So, we subtract 59 (which includes comedies) and 37 (which includes science fiction) from the total number of drama lovers: 196 - 59 - 37 = 100. Therefore, 100 customers liked only dramas.
c) To find the number of customers who liked exactly one type of movie, we need to add up the values in each individual circle (not including the overlapping regions). So, adding up the values within three circles: 196 + 153 + 88 = 437. But, since we've counted some movies more than once (in the overlapping regions), we need to subtract those values: 437 - 21 - 59 - 37 - 32 = 288. Thus, 288 customers liked exactly one type of movie.
In summary:
a) 201 liked none of these types of movies.
b) 100 liked only dramas.
c) 288 liked exactly one type of movie.