There are 30 students in a class. 20 of them play football, 16 play hockey and 16 play volleys. 9 play all three games, 15 play football and volley, 11 play football and hockey, while 10 play hockey and volley. a. Illustrate the information on a venn diagram
b. Using your venn diagram, find the number of students who play at least two games
c. How many students play exactly one game d. What is the probability that a student chosen at random from the class does not play any of the three games?
Let's first define the sets:
- F: Students who play football
- H: Students who play hockey
- V: Students who play volleyball
From the information provided, we can fill out the Venn diagram starting with the intersection of all three sets (students who play all three games), then the two-set intersections (students who play two of the games), and finally the unique players for each sport (students who play only one game).
We know:
- F ∩ H ∩ V = 9 (students who play all three games)
- F ∩ V = 15 (students who play football and volleyball)
- F ∩ H = 11 (students who play football and hockey)
- H ∩ V = 10 (students who play hockey and volleyball)
To find the number of students who play exactly two games, we need to subtract the number of students who play all three games from the counts of students playing any two games:
- (F ∩ V) ONLY = F ∩ V - F ∩ H ∩ V = 15 - 9 = 6
- (F ∩ H) ONLY = F ∩ H - F ∩ H ∩ V = 11 - 9 = 2
- (H ∩ V) ONLY = H ∩ V - F ∩ H ∩ V = 10 - 9 = 1
Now let's fill out the Venn diagram for the individual sets:
- F ONLY = Total football players - (F ∩ V) ONLY - (F ∩ H) ONLY - F ∩ H ∩ V = 20 - 6 - 2 - 9 = 3
- H ONLY = Total hockey players - (H ∩ V) ONLY - (F ∩ H) ONLY - F ∩ H ∩ V = 16 - 1 - 2 - 9 = 4
- V ONLY = Total volleyball players - (H ∩ V) ONLY - (F ∩ V) ONLY - F ∩ H ∩ V = 16 - 1 - 6 - 9 = 0
Now we can summarize what we know:
- F ONLY = 3
- H ONLY = 4
- V ONLY = 0
- (F ∩ V) ONLY = 6
- (F ∩ H) ONLY = 2
- (H ∩ V) ONLY = 1
- F ∩ H ∩ V = 9
Using this information, we can answer the questions:
a. The Venn Diagram cannot be illustrated in this text format, but if you were to create one on paper or with a drawing tool, each circle representing F, H, and V would overlap with each other, and you would place the numbers we calculated in the respective regions. In the center where all three circles overlap, you would have the number 9.
b. The number of students who play at least two games is the sum of the students in the two-set intersections:
(F ∩ V) ONLY + (F ∩ H) ONLY + (H ∩ V) ONLY + F ∩ H ∩ V = 6 + 2 + 1 + 9 = 18
c. The number of students who play exactly one game is the sum of the unique players for each set:
F ONLY + H ONLY + V ONLY = 3 + 4 + 0 = 7
d. The probability that a student chosen at random from the class does not play any game is found by subtracting the number of students who play at least one game from the total number of students, and then dividing by the total number of students:
Number who play no game = Total students - (F ONLY + H ONLY + V ONLY + (F ∩ V) ONLY + (F ∩ H) ONLY + (H ∩ V) ONLY + F ∩ H ∩ V)
= 30 - (3 + 4 + 0 + 6 + 2 + 1 + 9)
= 30 - 25
= 5
Probability (No game) = Number who play no game / Total students
= 5 / 30
= 1 / 6
= approximately 0.1667 (to four decimal places)
So, the probability that a student chosen at random does not play any of the three games is approximately 16.67%.