JEE Mathematics / Lakshmikanta Satapathy / Questions and Answers on probability part 8 which includes Bernoulli trials and Binomial distribution

1. Physics Helpline
L K Satapathy
Probability QA 8

2. Physics Helpline
L K Satapathy
QA Probability - 8
Q1 : An experiment succeeds thrice as often as it fails. Find the probability that in the
next 5 trials , there will be at least three successes.
3
4
p 
Ans : Given that there are 3 successes for each failure.
 Probability of success
11
4
q p  And Probability of failure
We are required to find the probability of at least 3 successes in 5 trials
 We have 5 Bernoulli trials  n = 5
We need to find P(X = 3) + P(X = 4) + P(X = 5)
Now, probability of x successes in n Bernoulli trials n n x x
xC q p


3. Physics Helpline
L K Satapathy
QA Probability - 8
 Required probability
5 2 3 5 4 5 5
3 4 5C q p C q p C p  
        
2 3 4 5
5 5 5
2 1 0
1 3 1 3 3
4 4 4 4 4
C C C  
        
2 3 4 5
5 4 1 3 1 3 35
1 2 4 4 4 4 4
  

27 81 24310 5
1024 1024 1024
    
270 405 243 918 459
1024 1024 512
[ ]Ans   

4. Physics Helpline
L K Satapathy
QA Probability - 8
Q2 : In an examination of five multiple choice questions , there are three possible
answers for each question. What is the probability that a candidate would get four or
more correct answers just by guessing ?
Ans : Guessing of the answers to multiple choice questions are Bernoulli trials
There are 5 multiple choice questions
 We have 5 Bernoulli trials  n = 5
There are 3 possible answers for each question out of which only 1 is correct
 Probability of success 1
3
p 
 Probability of failure 21
3
q p  
Let X represent the number of correct answers by guessing
 X is a random variable

5. Physics Helpline
L K Satapathy
QA Probability - 8
Now, probability of x successes in n Bernoulli trials is
( ) n n x x
xP X x C q p
 
 P (X = 4) + P (X = 5) 5 5 4 4 5 5 5 5
4 5C q p C q p 
 
   
4 5
2 1 15 1
3 3 3
    
5 5
10 1
3 3
 
5 4 5 5
1 0C q p C p 
11
243
[ ]Ans
We have to find the probability of 4 or more successes = P (X = 4) + P (X = 5)

6. Physics Helpline
L K Satapathy
QA Probability - 8
Q3 : A fair die is thrown six times. Find the probability of getting at most two sixes.
Ans : Successive throwing of a die are Bernoulli trials.
The die is thrown 6 times.
 We have 6 Bernoulli trials  n = 6
There is one six in the 6 possible outcomes in throwing a die once.
 Probability of success 1
6
p 
Let X represent the number of sixes obtained
 X is a random variable
 Probability of failure 51
6
q p  

7. Physics Helpline
L K Satapathy
QA Probability - 8
Now, probability of x successes in n Bernoulli trials is ( ) n n x x
xP X x C q p
 
We have to find the probability at most 2 successes = P (X = 0) + P (X = 1) + P (X = 2)
 P (X = 0) + P (X = 1) + P (X = 2)
6 6 6 5 6 4 2
0 1 2C q C q p C q p  
       
6 5 4 2
5 5 1 6 5 5 11 6
6 6 6 1 2 6 6
       

           
6 5 4 6 5 5
5 5 5 5 5 5 1 5
6 6 12 6 6 6 2 6
       
       
5 5
5 5 1 5 141
6 6 2 6 6
   
 
5
[7 5
6
]
3
Ans
1 56 , ,
6 6
n p q  We have obtained

8. Physics Helpline
L K Satapathy
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