## Important Probability questions set – Venn diagram

This question is frequently asked in the exams. Even in IBPS 2012 and SBI 2013, this question appeared with slight difference in amounts. Still most of the candidates are unable to solve this.
Question – There are 200 students in commerce batch of Khalsa college. Out of them 100 play cricket, 50 play hockey and 60 play basketball. 3o students play both cricket and hockey, 45 students play both cricket and basketball and 35 students play both hockey and basketball.
1) What is the maximum number of students who play all the games ?
2) What is minimum number of students who play all three games ?
3) What is the maximum number of students who play at least one game ?
4) What is the minimum number of students playing at least one game ?
Solution – To make the question simple, draw a Venn diagram and let x = number of students who play all games. Complete the diagram.

### Solution 1

What is the maximum value of x ?
Consider equation 30 – x, it states that maximum value of x = 30
In case x>30, for example 32 then 30 – x will be negative and number of students can’t be negative. So answer is 30.

### Solution 2

What is the minimum value of x ?
consider x-20 is possible as negative value of students is not possible. It is possible to deduct 20 from x.
So minimum value of x = 20

### Solution 3 and 4

For 3rd and 4th questions we will add everything :-
25 + x + x – 15 + x – 20 + 45 – x  + 35 – x + 30 – x + x =
= 100 + x
Here we add maximum and minimum value of x
so answer is 120 and 130 respectively.