JEE Mathematics / Lakshmikanta Satapathy / Questions and Answers on Conditional Probability which includes Problems on two dice and selecting questions from a question bank

1. Physics Helpline
L K Satapathy
Probability QA 4

2. Physics Helpline
L K Satapathy
Q1: A black die and a red die are rolled simultaneously. Find the conditional probability
of obtaining the sum 8 , given that the red die resulted in a number less than 4
QA Probability - 4
Ans : Let the events A = sum is 8  A = { 26 , 35 , 44 , 53 , 62 }
2 36( ) 2 1( )
( ) 18 36 1
[ ]
8 9
P A B
P A B s
P B
An

    
( ) 18( )
( ) 36
n B
P B
n S
 
( ) 2( )
( ) 36
n A B
P A B
n S

  
 AB = { 53 , 62 }  n(A) = 5 n(B) = 18 n(AB) = 2
 B = { 11, 12, 13, 21, 22, 23, 31, 32, 33, 41, 42, 43, 51, 52, 53, 61, 62, 63 }
B = number less than 4 on red die  any number from 1 to 3 on red die
and any number from 1 to 6 on black die.
For 2 dice, n(S) = 36 . We are required to find P(A/B)

3. Physics Helpline
L K Satapathy
Q2 : A fair die is rolled. Consider the events E = {1, 3, 5 } , F = {2, 3} and G = {2, 3, 4 , 5}.
QA Probability - 4
Ans : For rolling of 1 die , the sample space S = {1, 2, 3, 4, 5, 6}  n(S) = 6
Given : E = {1, 3, 5 } , F = {2, 3} and G = {2, 3, 4 , 5}
And EG = {3, 5} , FG = {2, 3} and E F G = {3}
 n( EG) = 2 , n(FG) = 2 and n(E F G) = 1
 n(E) = 3 , n(F) = 2 and n(G) = 4
Then find (i) P [(EF) /G] and (ii) P [(EF) /G]
( ) 2( )
( ) 6
n E G
P E G
n S

   
( ) 2( )
( ) 6
n F G
P F G
n S

  
( ) 1( )
( ) 6
n E F G
P E F G
n S
 
   
( ) 4( )
( ) 6
n G
P G
n S
 

4. Physics Helpline
L K Satapathy
QA Probability - 4
[( ) ] [( ) ( )]
( )
( ) ( )
P E F G P E G F G
P E F G
P G P G
    
  
( ) ( ) ( )
( )
P E G P F G P E F G
P G
     

2 2 1
36 6 6
4 4
[
6
]Ans
 
 
(i)
(ii) ( )
( )
( )
P E F G
P E F G
P G
 
 
1 6 1 [
4
]
6 4
Ans 

5. Physics Helpline
L K Satapathy
QA Probability - 4
Q3 : An instructor has a question bank consisting of 300 easy True/False questions ,
200 difficult True/False questions , 500 easy Multiple Choice questions and 400
difficult Multiple Choice questions. If a question is selected at random from the
question bank, then what is the probability that it will be an easy question given that
it is a multiple choice question.
Ans : Let the events ‘selecting an easy TFQ’ = A ,  n(A) = 300
We observe that A , B , C , and D are pair wise Disjoint events
The sample space S = all questions  n(S) = 300 +200 + 500 + 400 = 1400
‘selecting a difficult TFQ’ = B ,  n(B) = 200
‘selecting an easy MCQ’ = C ,  n(C) = 500
‘selecting a difficult MCQ’ = D ,  n(D) = 400
Let the event E = an easy Q = easy TFQ + easy MCQ n(E) = n(A) + n(C) = 800
and the event F = a MCQ = easy MCQ + difficult MCQ  n(F) = n(C) + n(D) = 900

6. Physics Helpline
L K Satapathy
QA Probability - 4
5 14( ) 5( )
( ) 9
[ ]
14 9
P E F
P E F Ans
P F

   
( ) 500 5( )
( ) 1400 14
n E F
P E F
n S

   
( ) 900 9( )
( ) 1400 14
n F
P F
n S
  
Now the event EF = easy MCQ n(EF) = 500
To find the probability of ‘an easy question , given that it is a MCQ’ = P(E/F)
For this , we need to find P(EF) and P(F)
To find P(E/F) , we will use the results n(S) = 1400 , n(F) = 900 and n(EF) = 500

7. Physics Helpline
L K Satapathy
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