JEE Mathematics / Lakshmikanta Satapathy / Problem on Probability of Mutually exclusive and Exhaustive events solved by axiomatic approach

1. Physics Helpline
L K Satapathy
Probability QA 3

2. Physics Helpline
L K Satapathy
Q1 : Two balls are drawn at random from a bag containing 2 white , 3
red , 5 green and 4 black balls , one by one , without replacement. Find
the probability that both the balls are of different colors.
QA Probability - 3
Ans : There are 2 white 3 red , 5 green and 4 black balls
 Total number of balls = 2 + 3 + 5 + 4 = 14
2 balls can be drawn from 14 balls in ways14
2C
14
2
14 13( ) 91
1 2
n S C    

Let the event ‘both are of same color’ = E
 the event ‘both are of different colors’ = E
E = Both are of the same color
 both white , both red , both green or both black
 P(E) = 1 – P(E)

3. Physics Helpline
L K Satapathy
QA Probability - 3
2 white balls can be drawn from 2 white balls in ways2
2C
2 red balls can be drawn from 3 red balls in ways3
2C
2 green balls can be drawn from 5 green balls in ways5
2C
2 black balls can be drawn from 4 black balls in ways4
2C
2 3 5 4
2 2 2 2( )n E C C C C    
3 2 5 4 4 31 1 3 10 6 20
1 2 1 2 1 2
          
  
( ) 20( )
( ) 91
n E
P E
n S
  
20 71( ) 1 ( ) 1
91
[ ]
91
P E P E Ans     

4. Physics Helpline
L K Satapathy
Q2 : If A , B and C are three mutually exclusive and exhaustive events
associated with a random experiment , then find P(A).
QA Probability - 2
Ans :
A , B and C are mutually exclusive  A  B = B  C = A  C = 
It is given that P(B) = P(A) and P(C) = P(B)3
2
1
2
and exhaustive  A  B  C = S
 P(ABC) = P(A) + P(B) + P(C) = P(S) = 1 . . . (1)
3 3 1 3( ) ( ) ( ) ( ) ( )
2 2 2 4
Let P A p P B P A p and P C P B p     
3 3 4 6 3(1) 1 1
2 4 4
p p p p      
4( )
13
[ ]AnsP A p  

5. Physics Helpline
L K Satapathy
Q3 : Two dice are thrown together. What is the probability that the sum
of the numbers on the two faces is neither divisible by 3 nor by 4?
QA Probability - 3
Ans : Random experiment : Throwing of 2 dice.  n(S) = 36
Let the event ‘sum divisible by 3’ = A
And the event ‘sum divisible by 4’ = B
We are required to find P(A B) = P(A B) = 1 – P(A  B)
Now A = {12 , 15 , 21 , 24 , 33 , 36 , 42 , 45 , 51 , 54 , 63 , 66}  n(A) = 12
And B = {13 , 22 , 26 , 31 , 35 , 44 , 53 , 62 , 66}  n(B) = 9
 A  B = { 66 }  n(A  B) = 1

6. Physics Helpline
L K Satapathy
QA Probability - 3
( ) 12 1( )
( ) 36 3
n A
P A
n S
   
( ) 9 1( )
( ) 36 4
n B
P B
n S
  
( ) 1( )
( ) 36
n A B
P A B
n S

  
( ) ( ) ( ) ( )P A B P A P B P A B     
1 1 1 12 9 1 20 5
3 4 36 36 36 9
      
5 4( ) 1 ( ) 1
9 9
[ ]P A B P A B Ans       

7. Physics Helpline
L K Satapathy
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