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?
To answer these questions, we will use a Venn diagram and the principle of inclusion-exclusion to calculate the desired quantities.
**Venn Diagram:**
A Venn diagram is a visual representation of the relationships between different sets. Since we are dealing with three sports - football (F), hockey (H), and volley (V) - our Venn diagram will have three overlapping circles.
1. We start by placing the 9 students who play all three sports in the center intersection of all three circles.
2. Now consider the students who play football and volley (F ∩ V). There are 15 in total, but we have already accounted for the 9 who also play hockey. So, we place 15 - 9 = 6 in the Football-Volley overlap, excluding the hockey circle.
3. Next, for the students who play football and hockey (F ∩ H), we again subtract the 9 who play all three games: 11 - 9 = 2 are in the Football-Hockey overlap (excluding volley).
4. For the hockey and volley players (H ∩ V), there are 10 in total, but minus the 9 who play all three: 10 - 9 = 1 goes in the Hockey-Volley overlap, excluding football.
5. Now we fill in the number of students who play only one sport. We subtract those in the two-game and three-game intersections from the total for each sport:
- For football (F), we have 20 total, but we need to subtract those who play hockey (2) and volley (6) and all three (9): 20 - (2 + 6 + 9) = 20 - 17 = 3 play only football.
- For hockey (H), there are 16 total, subtracting those who play football (2), volley (1), and all three (9): 16 - (2 + 1 + 9) = 16 - 12 = 4 play only hockey.
- For volley (V), again 16 total, subtract those who play football (6), hockey (1), and all three (9): 16 - (6 + 1 + 9) = 16 - 16 = 0, so no one plays only volley.
**Answers:**
**b. Number of students who play at least two games:**
This is the sum of those who play exactly two games plus those who play all three games.
= (6 Football-Volley only) + (2 Football-Hockey only) + (1 Hockey-Volley only) + (9 All three games)
= 6 + 2 + 1 + 9
= 18 students play at least two games.
**c. Number of students who play exactly one game:**
This is the sum of those who play only one sport:
= (3 Football only) + (4 Hockey only) + (0 Volley only)
= 3 + 4 + 0
= 7 students play exactly one game.
**d. Probability that a student chosen at random does not play any of the three games:**
To find the number of students who don't play any game, we subtract all those who play at least one game from the total number of students.
= 30 total students - (3 Football only + 4 Hockey only + 6 Football-Volley only + 2 Football-Hockey only + 1 Hockey-Volley only + 9 All three games)
= 30 - (3 + 4 + 6 + 2 + 1 + 9)
= 30 - 25
= 5 students do not play any of the three games.
Now, the probability is the number of students who don't play any game divided by the total number of students.
= 5 students do not play / 30 total students
= 1/6 or approximately 0.1667.
Thus, there is approximately a 16.67% chance that a student chosen at random does not play any of the games.