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NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
Exercise 13.3                                                                        Page No: 555 
1. An urn contains 5 red and 5 black balls. A ball is drawn at random; its colour is noted and is returned to the 
urn. Moreover, 2 additional balls of the colour drawn are put in the urn, and then a ball is drawn at random. 
What is the probability that the second ball is red? 
Solution: 
Given, the urn contains 5 red and 5 black balls. 
Let in the first attempt, the ball drawn is of red colour. 
? P (probability of drawing a red ball) = 5/10 = ½ 
Now, the two balls of the same colour (red) are added to the urn then the urn contains 7 red and 5 black balls. 
? P (probability of drawing a red ball) = 7/12 
Now, let in the first attempt, the ball drawn is of black colour. 
? P (probability of drawing a black ball) = 5/10 = ½ 
Now, the two balls of the same colour (black) are added to the urn then the urn contains 5 red and 7 black balls. 
? P (probability of drawing a red ball) = 5/12 
Therefore, the probability of drawing the second ball as of red colour is 
 
2. A bag contains 4 red and 4 black balls, and another bag contains 2 red and 6 black balls. One of the two bags 
is selected at random, and a ball is drawn from the bag, which is found to be red. Find the probability that the 
ball is drawn from the first bag. 
Solution: 
Let E 1 be the event of choosing bag I, E 2 be the event of choosing the bag, say bag II and A be the event of drawing a 
red ball. 
Then, P (E 1) = P (E 2) = 1/2 
Also, P (A|E 1) = P (drawing a red ball from bag I) = 4/8 = ½ 
And P (A|E 2) = P (drawing a red ball from bag II) = 2/8 = ¼ 
Now, the probability of drawing a ball from bag I, being given that it is red, is P (E 1|A). 
By using Bayes’ theorem, we have 
Page 2


 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
Exercise 13.3                                                                        Page No: 555 
1. An urn contains 5 red and 5 black balls. A ball is drawn at random; its colour is noted and is returned to the 
urn. Moreover, 2 additional balls of the colour drawn are put in the urn, and then a ball is drawn at random. 
What is the probability that the second ball is red? 
Solution: 
Given, the urn contains 5 red and 5 black balls. 
Let in the first attempt, the ball drawn is of red colour. 
? P (probability of drawing a red ball) = 5/10 = ½ 
Now, the two balls of the same colour (red) are added to the urn then the urn contains 7 red and 5 black balls. 
? P (probability of drawing a red ball) = 7/12 
Now, let in the first attempt, the ball drawn is of black colour. 
? P (probability of drawing a black ball) = 5/10 = ½ 
Now, the two balls of the same colour (black) are added to the urn then the urn contains 5 red and 7 black balls. 
? P (probability of drawing a red ball) = 5/12 
Therefore, the probability of drawing the second ball as of red colour is 
 
2. A bag contains 4 red and 4 black balls, and another bag contains 2 red and 6 black balls. One of the two bags 
is selected at random, and a ball is drawn from the bag, which is found to be red. Find the probability that the 
ball is drawn from the first bag. 
Solution: 
Let E 1 be the event of choosing bag I, E 2 be the event of choosing the bag, say bag II and A be the event of drawing a 
red ball. 
Then, P (E 1) = P (E 2) = 1/2 
Also, P (A|E 1) = P (drawing a red ball from bag I) = 4/8 = ½ 
And P (A|E 2) = P (drawing a red ball from bag II) = 2/8 = ¼ 
Now, the probability of drawing a ball from bag I, being given that it is red, is P (E 1|A). 
By using Bayes’ theorem, we have 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
 
3. Of the students in a college, it is known that 60% reside in the hostel, and 40% are day scholars (not residing 
in the hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% 
of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at 
random from the college, and he has an A grade; what is the probability that the student is a hostlier? 
Solution: 
Page 3


 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
Exercise 13.3                                                                        Page No: 555 
1. An urn contains 5 red and 5 black balls. A ball is drawn at random; its colour is noted and is returned to the 
urn. Moreover, 2 additional balls of the colour drawn are put in the urn, and then a ball is drawn at random. 
What is the probability that the second ball is red? 
Solution: 
Given, the urn contains 5 red and 5 black balls. 
Let in the first attempt, the ball drawn is of red colour. 
? P (probability of drawing a red ball) = 5/10 = ½ 
Now, the two balls of the same colour (red) are added to the urn then the urn contains 7 red and 5 black balls. 
? P (probability of drawing a red ball) = 7/12 
Now, let in the first attempt, the ball drawn is of black colour. 
? P (probability of drawing a black ball) = 5/10 = ½ 
Now, the two balls of the same colour (black) are added to the urn then the urn contains 5 red and 7 black balls. 
? P (probability of drawing a red ball) = 5/12 
Therefore, the probability of drawing the second ball as of red colour is 
 
2. A bag contains 4 red and 4 black balls, and another bag contains 2 red and 6 black balls. One of the two bags 
is selected at random, and a ball is drawn from the bag, which is found to be red. Find the probability that the 
ball is drawn from the first bag. 
Solution: 
Let E 1 be the event of choosing bag I, E 2 be the event of choosing the bag, say bag II and A be the event of drawing a 
red ball. 
Then, P (E 1) = P (E 2) = 1/2 
Also, P (A|E 1) = P (drawing a red ball from bag I) = 4/8 = ½ 
And P (A|E 2) = P (drawing a red ball from bag II) = 2/8 = ¼ 
Now, the probability of drawing a ball from bag I, being given that it is red, is P (E 1|A). 
By using Bayes’ theorem, we have 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
 
3. Of the students in a college, it is known that 60% reside in the hostel, and 40% are day scholars (not residing 
in the hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% 
of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at 
random from the college, and he has an A grade; what is the probability that the student is a hostlier? 
Solution: 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
 
 
4. In answering a question on a multiple-choice test, a student either knows the answer or guesses. Let 3/4 be the 
probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who 
guesses the answer will be correct with probability 1/4. What is the probability that the student knows the 
answer, given that he answered it correctly? 
Page 4


 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
Exercise 13.3                                                                        Page No: 555 
1. An urn contains 5 red and 5 black balls. A ball is drawn at random; its colour is noted and is returned to the 
urn. Moreover, 2 additional balls of the colour drawn are put in the urn, and then a ball is drawn at random. 
What is the probability that the second ball is red? 
Solution: 
Given, the urn contains 5 red and 5 black balls. 
Let in the first attempt, the ball drawn is of red colour. 
? P (probability of drawing a red ball) = 5/10 = ½ 
Now, the two balls of the same colour (red) are added to the urn then the urn contains 7 red and 5 black balls. 
? P (probability of drawing a red ball) = 7/12 
Now, let in the first attempt, the ball drawn is of black colour. 
? P (probability of drawing a black ball) = 5/10 = ½ 
Now, the two balls of the same colour (black) are added to the urn then the urn contains 5 red and 7 black balls. 
? P (probability of drawing a red ball) = 5/12 
Therefore, the probability of drawing the second ball as of red colour is 
 
2. A bag contains 4 red and 4 black balls, and another bag contains 2 red and 6 black balls. One of the two bags 
is selected at random, and a ball is drawn from the bag, which is found to be red. Find the probability that the 
ball is drawn from the first bag. 
Solution: 
Let E 1 be the event of choosing bag I, E 2 be the event of choosing the bag, say bag II and A be the event of drawing a 
red ball. 
Then, P (E 1) = P (E 2) = 1/2 
Also, P (A|E 1) = P (drawing a red ball from bag I) = 4/8 = ½ 
And P (A|E 2) = P (drawing a red ball from bag II) = 2/8 = ¼ 
Now, the probability of drawing a ball from bag I, being given that it is red, is P (E 1|A). 
By using Bayes’ theorem, we have 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
 
3. Of the students in a college, it is known that 60% reside in the hostel, and 40% are day scholars (not residing 
in the hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% 
of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at 
random from the college, and he has an A grade; what is the probability that the student is a hostlier? 
Solution: 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
 
 
4. In answering a question on a multiple-choice test, a student either knows the answer or guesses. Let 3/4 be the 
probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who 
guesses the answer will be correct with probability 1/4. What is the probability that the student knows the 
answer, given that he answered it correctly? 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
Solution: 
Let E 1 be the event that the student knows the answer, E 2 be the event that the student guesses the answer, and A be the 
event that the answer is correct. 
Then, P (E 1) = ¾ 
And P (E 2) = ¼ 
Also, P (A|E 1) = P (correct answer given that he knows) = 1 
And P (A|E 2) = P (correct answer given that he guesses) = ¼ 
Now, the probability that he knows the answer, being given that answer is correct, is P (E 1|A). 
By using Bayes’ theorem, we have 
 
 
5. A laboratory blood test is 99% effective in detecting a certain disease when it is, in fact, present. However, the 
test also yields a false positive result for 0.5% of the healthy person tested (i.e., if a healthy person is tested, then, 
with a probability 0.005, the test will imply he has the disease). If 0.1 per cent of the population actually has the 
disease, what is the probability that a person has the disease given that his test result is positive? 
Solution: 
Let E 1 be the event that person has a disease, E 2 be the event that the person does not have a disease, and A be the event 
that the blood test is positive. 
Page 5


 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
Exercise 13.3                                                                        Page No: 555 
1. An urn contains 5 red and 5 black balls. A ball is drawn at random; its colour is noted and is returned to the 
urn. Moreover, 2 additional balls of the colour drawn are put in the urn, and then a ball is drawn at random. 
What is the probability that the second ball is red? 
Solution: 
Given, the urn contains 5 red and 5 black balls. 
Let in the first attempt, the ball drawn is of red colour. 
? P (probability of drawing a red ball) = 5/10 = ½ 
Now, the two balls of the same colour (red) are added to the urn then the urn contains 7 red and 5 black balls. 
? P (probability of drawing a red ball) = 7/12 
Now, let in the first attempt, the ball drawn is of black colour. 
? P (probability of drawing a black ball) = 5/10 = ½ 
Now, the two balls of the same colour (black) are added to the urn then the urn contains 5 red and 7 black balls. 
? P (probability of drawing a red ball) = 5/12 
Therefore, the probability of drawing the second ball as of red colour is 
 
2. A bag contains 4 red and 4 black balls, and another bag contains 2 red and 6 black balls. One of the two bags 
is selected at random, and a ball is drawn from the bag, which is found to be red. Find the probability that the 
ball is drawn from the first bag. 
Solution: 
Let E 1 be the event of choosing bag I, E 2 be the event of choosing the bag, say bag II and A be the event of drawing a 
red ball. 
Then, P (E 1) = P (E 2) = 1/2 
Also, P (A|E 1) = P (drawing a red ball from bag I) = 4/8 = ½ 
And P (A|E 2) = P (drawing a red ball from bag II) = 2/8 = ¼ 
Now, the probability of drawing a ball from bag I, being given that it is red, is P (E 1|A). 
By using Bayes’ theorem, we have 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
 
3. Of the students in a college, it is known that 60% reside in the hostel, and 40% are day scholars (not residing 
in the hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% 
of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at 
random from the college, and he has an A grade; what is the probability that the student is a hostlier? 
Solution: 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
 
 
4. In answering a question on a multiple-choice test, a student either knows the answer or guesses. Let 3/4 be the 
probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who 
guesses the answer will be correct with probability 1/4. What is the probability that the student knows the 
answer, given that he answered it correctly? 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
Solution: 
Let E 1 be the event that the student knows the answer, E 2 be the event that the student guesses the answer, and A be the 
event that the answer is correct. 
Then, P (E 1) = ¾ 
And P (E 2) = ¼ 
Also, P (A|E 1) = P (correct answer given that he knows) = 1 
And P (A|E 2) = P (correct answer given that he guesses) = ¼ 
Now, the probability that he knows the answer, being given that answer is correct, is P (E 1|A). 
By using Bayes’ theorem, we have 
 
 
5. A laboratory blood test is 99% effective in detecting a certain disease when it is, in fact, present. However, the 
test also yields a false positive result for 0.5% of the healthy person tested (i.e., if a healthy person is tested, then, 
with a probability 0.005, the test will imply he has the disease). If 0.1 per cent of the population actually has the 
disease, what is the probability that a person has the disease given that his test result is positive? 
Solution: 
Let E 1 be the event that person has a disease, E 2 be the event that the person does not have a disease, and A be the event 
that the blood test is positive. 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 13 – 
Probability  
As E 1 and E 2 are the events which are complimentary to each other. 
Then, P (E 1) + P (E 2) = 1 
? P (E 2) = 1 – P (E 1) 
Then, P (E 1) = 0.1% = 0.1/100 = 0.001 and P (E 2) = 1 – 0.001 = 0.999 
Also, P (A|E 1) = P (result is positive given that person has disease) = 99% = 0.99 
And P (A|E 2) = P (result is positive given that person has no disease) = 0.5% = 0.005 
Now, the probability that a person has a disease, given that his test result is positive, is P (E 1|A). 
By using Bayes’ theorem, we have

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