Question

There are 30 students in a class. 20 of them play football, 16 play hockey and 16 play volley. 9 play all the three games, 15 play football and volley, 11 play football and hockey, while 10 play hockey and volley. a) Ilustrate the information on a Venn diagram b) Using your Venn diagram, find the number of students who play at least two gamnes c) What is probability that a student chosen at random from the class does not play any of the three games?​

Answer 1

a) Here's a Venn diagram to illustrate the information:

```
F--15--(3)--9--(6)
/ \
/ \
/ \
H--11--(2)--9--(1) V--10--(2)--9--(1)
\ /
\ /
\ /
(5)----16----(2)
```

The numbers in parentheses represent the number of students who play all the combinations of the different games.

b) To find the number of students who play at least two games, we need to add up the number of students in the intersections of the circles. From the Venn diagram, we can see that there are three intersections:

- Football and Hockey: 11 students
- Football and Volley: 15 students
- Hockey and Volley: 10 students

Adding up these numbers, we get:

11 + 15 + 10 = 36

Therefore, 36 students play at least two games.

c) To find the probability that a student chosen at random from the class does not play any of the three games, we need to find the number of students who do not play any of the games and divide that by the total number of students. From the Venn diagram, we can see that the number of students who do not play any of the games is:

30 - (20 + 16 + 16 - 9 -15 - 11 + 10) = 30 - 6 = 24

Therefore, the probability that a student chosen at random from the class does not play any of the three games is:

24/30 = 4/5 or 0.8 (as a decimal)