There are 30 student in a class 20 of them play football 16 play hockey and 16 play volley. 9 play all the 3 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) what is the probability that a student choosen at the random from the class does not play any of the 3 games.
To answer this question, we can use a Venn diagram to represent the information given.
(a) Let's create a Venn diagram with three overlapping circles representing football (F), hockey (H), and volley (V).
Since we know that there are 30 students in the class, we can write this number outside the circles.
Next, we'll fill in the information we have.
- 20 students play football, so we write 20 inside the football circle (F).
- 16 students play hockey, so we write 16 inside the hockey circle (H).
- 16 students play volley, so we write 16 inside the volley circle (V).
Next, we know that 9 students play all three games, which means this number goes in the overlapping region of all three circles.
Then, we know that 15 students play football and volley, so we put this number in the overlapping region between the football and volley circles (FV).
Similarly, 11 students play football and hockey, so this number goes in the overlapping region between the football and hockey circles (FH).
Lastly, 10 students play hockey and volley, so we write this number in the overlapping region between the hockey and volley circles (HV).
To find the remaining numbers, we can use the principle of inclusion-exclusion.
We add up the numbers in each circle:
- F: 20
- H: 16
- V: 16
Then we subtract the sums of each overlap region:
- FV: 15
- FH: 11
- HV: 10
Finally, we add the number of students who play all three games:
- All three circles (FHV): 9
Now, let's fill in the numbers on the Venn diagram:
F
/ \
/ \
/ \
FH FHV HV
\ /
\ /
\ /
H
Then, let's fill in the numbers:
- F: 20
- H: 16
- V: 16
- FV: 15
- FH: 11
- HV: 10
- FHV: 9
(b) The number of students who play at least two games can be calculated by adding the numbers in the overlap regions (FV, FH, HV, and FHV).
So, the number of students who play at least two games is 15 + 11 + 10 + 9 = 45.
(c) The probability that a student chosen at random from the class does not play any of the three games is equal to the number of students who do not play any games divided by the total number of students (30).
To calculate the number of students who do not play any games, we can subtract the sum of all the numbers inside the circles (F, H, and V), the number of students who play at least two games (45), and the number of students who play all three games (9), from the total number of students (30).
So, the number of students who do not play any games is 30 - F - H - V - (FV + FH + HV) - FHV = 30 - 20 - 16 - 16 - (15 + 11 + 10) - 9 = 30 - 72 = -42.
However, it doesn't make sense to have a negative number of students, so it seems there is an error in the initial data or calculations. Please double-check the information provided.