In a school, 27 students were asked their preference for three brands of soft drink: Fanta, Coca-Cola, and Sprite. 15 liked Sprite, 16 liked Fanta, and 5 liked all three. 12 prefer... In a school, 27 students were asked their preference for three brands of soft drink: Fanta, Coca-Cola, and Sprite. 15 liked Sprite, 16 liked Fanta, and 5 liked all three. 12 preferred Coca-Cola and Fanta, 6 preferred Coca-Cola and Sprite, and 6 preferred Sprite only. Illustrate the information on a Venn diagram. (a) Find how many students liked: (i) Coca-Cola, (ii) Fanta or Sprite but not Coca-Cola, (iii) Fanta and Sprite but not Coca-Cola, (iv) only one brand, (v) only two brands. (b) Find the probability that a student selected liked only Fanta.
Understand the Problem
The question is asking us to find the number of students who liked certain combinations of Fanta, Coca-Cola, and Sprite, given data about student preferences. We are asked to illustrate this information on a Venn diagram and then answer questions about the number of students who liked specific combinations of the drinks. Finally, we need to calculate the probability that a randomly selected student liked only Fanta.
Answer
(i) 17 (ii) 10 (iii) 3 (iv) 11 (v) 11 (b) $\frac{1}{27}$
Answer for screen readers
(i) 17 students liked Coca-Cola. (ii) 10 students liked Fanta or Sprite but not Coca-Cola. (iii) 3 students liked Fanta and Sprite not Coca-Cola. (iv) 11 students liked only one brand. (v) 11 students liked only two brands. (b) The probability that a student selected liked only Fanta is $\frac{1}{27}$.
Steps to Solve
- Draw the Venn Diagram
Draw a Venn diagram with three overlapping circles representing Fanta, Coca-Cola, and Sprite.
- Fill in the intersection of all three circles
5 students liked all three drinks, so place "5" in the intersection of all three circles.
- Calculate the number of students who liked Coca-Cola and Fanta only
12 students liked Coca-Cola and Fanta, and 5 liked all three. Therefore, $12 - 5 = 7$ liked only Coca-Cola and Fanta. Place "7" in the intersection of Coca-Cola and Fanta, outside the Sprite circle.
- Calculate the number of students who liked Coca-Cola and Sprite only
6 students liked Coca-Cola and Sprite, and 5 liked all three. Therefore, $6 - 5 = 1$ liked only Coca-Cola and Sprite. Place "1" in the intersection of Coca-Cola and Sprite, outside the Fanta circle.
- Calculate the number of students who liked only Sprite
6 students liked Sprite only. Place "6" in the Sprite circle, outside of the other circles' intersections.
- Calculate the number of students who liked Sprite and Fanta only
We know that 15 students liked Sprite in total. We also know that:
- 6 liked only Sprite
- 1 liked Coca-Cola and Sprite only
- 5 liked all three
So, the number of students who liked Sprite and Fanta only is: $15 - 6 - 1 - 5 = 3$. Place "3" in the intersection of Fanta and Sprite, outside the Coca-Cola circle.
- Calculate the number of students who liked only Fanta
We know that 16 students liked Fanta in total. We also know that:
- 7 liked Fanta and Coca-Cola only
- 3 liked Fanta and Sprite only
- 5 liked all three
So, the number of students who liked only Fanta is: $16 - 7 - 3 - 5 = 1$. Place "1" in the Fanta circle, outside of the other circles' intersections.
- Calculate the number of students who liked only Coca-cola
We know that there are 27 students in total. We can sum all the values in our Venn diagram so far: $1 + 6 + 7 + 1 + 3 + 5 = 23$. Thus, the number of students who liked only Coca-cola is $27 - 23 = 4$.
- Determine the number of students who liked Coca-Cola
The number of students who liked Coca-Cola is the sum of those who liked only Coca-Cola, Coca-Cola and Fanta, Coca-Cola and Sprite, and all three: $4 + 7 + 1 + 5 = 17$.
- Determine the number of students who liked Fanta or Sprite but not Coca-Cola
This is the sum of those who liked only Fanta, only Sprite, and Fanta and Sprite only: $1 + 6 + 3 = 10$.
- Determine the number of students who liked Fanta and Sprite, but not Coca-cola
This is the number of students who liked only Fanta and Sprite, which is 3.
- Determine the number of students who liked only one brand
This is the sum of those who liked only Fanta, only Coca-Cola, and only Sprite: $1 + 4 + 6 = 11$.
- Determine the number of students who liked only two brands
This is the sum of those who liked only Fanta and Coca-Cola, only Fanta and Sprite, and only Coca-Cola and Sprite: $7 + 3 + 1 = 11$.
- Calculate the probability that a student selected liked only Fanta
The number of students who liked only Fanta is 1, and the total number of students is 27. Therefore, the probability is $\frac{1}{27}$.
(i) 17 students liked Coca-Cola. (ii) 10 students liked Fanta or Sprite but not Coca-Cola. (iii) 3 students liked Fanta and Sprite not Coca-Cola. (iv) 11 students liked only one brand. (v) 11 students liked only two brands. (b) The probability that a student selected liked only Fanta is $\frac{1}{27}$.
More Information
Venn diagrams are great for visually breaking down data!
Tips
A common mistake is to forget to subtract the intersection of all three sets when calculating the number of students who like only two brands. Also, be sure to double-check all your additions and subtractions to minimize arithmetic errors, and make sure all sections of the Venn diagram make logical sense.
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