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Probability is not a certainty, but it is the possibility with a heartbeat. In general terms, probability is how likely something is to happen. It is like a number between 0 and 1. If it is 0, it will not happen, but if it is 1, it will definitely happen. Anything in between 0 and 1 means it might happen. In the Probability Class 12 NCERT solutions, students will learn concepts of probability distributions, random variables, Bayes' theorem, and conditional probability thoroughly. The main goal of these solutions by NCERT is to prepare students for the Class 12 board exam as well as other competitive exams.
Every day before leaving our houses, we check for weather forecasts. Ever wondered how metrologists predict the weather? Ever thought about how these clothing brands determine what’s going to trend or on what models, and how AI and machine learning work? The solution is in the Probability chapter! Experienced Careers360 teachers abiding by the latest CBSE syllabus have curated these class 12 Maths NCERT solutions of Probability. Students can additionally refer to the NCERT Exemplar Solutions for Class 12 Maths Chapter 13 Probability for better practice and understanding of the topic.
Students who wish to access the Class 12 Maths Chapter 13 NCERT Solutions can click on the link below to download the complete solution in PDF.
Class 12 Maths chapter 13 solutions - Exercise: 13.1 Page number: 413-415 Total questions: 17 |
Question 1: Given that
Answer:
It is given that
Question 2: Compute
Answer:
It is given that
Question 3: If
Answer:
It is given that P(A)=0.8, P(B)=0.5 and P(B|A)=0.4
Question 3: If
Answer:
It is given that
Question 3: If
Answer:
It is given that
Question 4: Evaluate
Answer:
Given in the question
We know that:
Use,
Question 5: If
Answer:
Given in the question
By using the formula:
Question 5: If
Answer:
It is given that -
We know that:
Question 5: If
Answer:
Given in the question-
Use formula
Question 6: A coin is tossed three times, where
(i)E : head on third toss ,F : heads on first two tosses
Answer:
The sample space S when a coin is tossed three times is
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
It can be seen that the sample space (S) has 8 elements.
Total number of outcomes
According to the question
E: head on third toss, F: heads on first two tosses
Question 6: A coin is tossed three times, where
(ii)E : at least two heads ,F : at most two heads
Answer:
The sample space S when a coin is tossed three times is
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
It can be seen that the sample space (S) has 8 elements.
Total number of outcomes
According to question
E : at least two heads , F : at most two heads
Question 6: A coin is tossed three times, where
(iii)E : at most two tails ,F : at least one tail
Answer:
The sample space S when a coin is tossed three times is
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
It can be seen that the sample space (S) has 8 elements.
Total number of outcomes
According to the question
E: at most two tails, F: at least one tail
Question 7: Two coins are tossed once, where
(i) E : tail appears on one coin, F : one coin shows head
Answer:
E : tail appears on one coin, F : one coin shows head
Total outcomes =4
Question 7: Two coins are tossed once, where
(ii)E : no tail appears,F : no head appears
Answer:
E : no tail appears, F : no head appears
Total outcomes =4
Question 8: A die is thrown three times,
E : 4 appears on the third toss, F : 6 and 5 appears respectively on first two tosses
Answer:
E : 4 appears on the third toss, F : 6 and 5 appears respectively on first two tosses
Total outcomes
Question 9: Mother, father and son line up at random for a family picture
E : son on one end, F : father in middle
Answer:
E : son on one end, F : father in middle
Total outcomes
Let S be son, M be mother and F be father.
Then we have,
Question 10: A black and a red dice are rolled.
Answer:
A black and a red dice are rolled.
Total outcomes
Let the A be event obtaining a sum greater than
Question 10: A black and a red dice are rolled.
Answer:
A black and a red dice are rolled.
Total outcomes
Let A be the event obtaining a sum 8 and B be a event that the red die resulted in a number less than
Red dice is rolled after black dice.
Question 11: A fair die is rolled. Consider events
Answer:
A fair die is rolled.
Total oucomes
Question 11: A fair die is rolled. Consider events
Answer:
A fair die is rolled.
Total outcomes
Question 11: A fair die is rolled. Consider events
Answer:
A fair die is rolled.
Total outcomes
Answer:
Assume that each born child is equally likely to be a boy or a girl.
Let first and second girl are denoted by
If a family has two children, then total outcomes
Let A= both are girls
and B= the youngest is a girl =
Therefore, the required probability is
Question 12: Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that
Answer:
Assume that each born child is equally likely to be a boy or a girl.
Let first and second girl are denoted by
If a family has two children, then total outcomes
Let A= both are girls
and C= at least one is a girl =
Answer:
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.
Total number of questions
Let A = question be easy.
Let B = multiple-choice question
Therefore, the required probability is
Answer:
Two dice are thrown.
Total outcomes
Let A be the event ‘the sum of numbers on the dice is 4.
Let B be the event that two numbers appearing on throwing two dice are different.
Therefore, the required probability is
Answer:
Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin.
Total outcomes
Total number of outcomes =20
Let A be a event when coin shows a tail.
Let B be a event that at least one die shows a 3’.
Question 16: In the following Exercise 16 choose the correct answer:
Answer:
It is given that
Hence,
Thus, the correct option is C.
Question 17: In the following Exercise 17 choose the correct answer:
If
Answer:
It is given that
Hence, option D is correct.
Class 12 Maths chapter 13 solutions - Exercise: 13.2 Page number: 421-423 Total questions: 18 |
Question 1: If
Answer:
Given:
So we have,
Answer:
Two cards are drawn at random and without replacement from a pack of 52 playing cards.
There are 26 black cards in a pack of 52.
Let
Then, we have
Let
Then, we have
The probability that both the cards are black
Answer:
Total oranges = 15
Good oranges = 12
Bad oranges = 3
Let
The, we have
Let
Let
The probability that a box will be approved for sale
Answer:
A fair coin and an unbiased die are tossed, then the total outputs are:
A is the event ‘the head appears on the coin’.
Total outcomes of A are :
B is the event ‘3 on the die’.
Total outcomes of B are :
Also,
Hence, A and B are independent events.
Answer:
Total outcomes
Outcomes of A
Outcomes of B
Also,
Thus, both events A and B are not independent.
Question 6: Let
Answer:
Given :
For events E and F to be independent, we need
Hence, E and F are not independent events.
Question 7: Given that the events
Answer:
Given,
Also, A and B are mutually exclusive means
Question 7: Given that the events
Answer:
Given,
Also, A and B are independent events means
Question 8: Let A and B be independent events with
Find
Answer:
Given: A and B are independent events
So, we have
Question 8: Let
Answer:
Given: A and B are independent events
So, we have
We have,
Question 8: Let
Answer:
Given: A and B are independent events
So, we have
Question 8: Let A and B be independent events with
Answer:
Given: A and B are independent events
So, we have
Question 9: If
Answer:
If
Question 10: Events A and B are such that
Answer:
If
As we can see
Hence, A and B are not independent.
Question 11: Given two independent events
Answer:
Given two independent events
Also, we know
Question 11: Given two independent events A and B such that
Answer:
Given two independent events
Question 11: Given two independent events A and B such that
Answer:
Question 11: Given two independent events
Answer:
Question 12: A die is tossed thrice. Find the probability of getting an odd number at least once.
Answer:
A die is tossed thrice.
Outcomes
Odd numbers
The probability of getting an odd number on the first throw
The probability of getting an even number
Probability of getting even number three times
The probability of getting an odd number at least once
= 1 - the probability of getting an odd number in none of the throws
= 1 - probability of getting an even number three times
Answer:
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls.
Total balls =18
Black balls = 10
Red balls = 8
The probability of getting a red ball in the first draw
The ball is replaced after drawing the first ball.
The probability of getting a red ball in the second draw
The probability that both balls are red
Question 13: Two balls are drawn at random with replacement from a box containing
(ii) first ball is black and second is red.
Answer:
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls.
Total balls =18
Black balls = 10
Red balls = 8
The probability of getting a black ball in the first draw
The ball is replaced after drawing the first ball.
The probability of getting a red ball in the second draw
The probability that the first ball is black and the second is red
Question 13: Two balls are drawn at random with replacement from a box containing
(iii) one of them is black and other is red.
Answer:
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls.
Total balls =18
Black balls = 10
Red balls = 8
Let the first ball is black and the second ball is red.
The probability of getting a black ball in the first draw
The ball is replaced after drawing the first ball.
The probability of getting a red ball in the second draw
The probability that the first ball is black and the second is red
Let the first ball is red and the second ball is black.
The probability of getting a red ball in the first draw
The probability of getting a black ball in the second draw
the probability that the first ball is red and the second is black
Thus,
The probability that one of them is black and the other is red = the probability that the first ball is black and the second is red + the probability that the first ball is red and the second is black
Answer:
Since the problem is solved independently by A and B,
probability that the problem is solved
(ii) exactly one of them solves the problem
Answer:
probability that exactly one of them solves the problem
probability that exactly one of them solves the problem
(i) E : ‘the card drawn is a spade’
F : ‘the card drawn is an ace’
Answer:
One card is drawn at random from a well-shuffled deck of
Total ace = 4
total spades =13
E : ‘the card drawn is a spade
F : ‘the card drawn is an ace’
Hence, E and F are independent events.
F : ‘the card drawn is a king’
Answer:
One card is drawn at random from a well-shuffled deck of
Total black card = 26
total king =4
E : ‘the card drawn is black’
F : ‘the card drawn is a king’
Hence, E and F are independent events.
(iii) E : ‘the card drawn is a king or queen’
F : ‘the card drawn is a queen or jack’.
Answer:
One card is drawn at random from a well-shuffled deck of
Total king or queen = 8
total queen or jack = 8
E : ‘the card drawn is a king or queen’
F : ‘the card drawn is a queen or jack’.
Hence, E and F are not independent events
(a) Find the probability that she reads neither Hindi nor English newspapers
Answer:
H :
E :
The probability that she reads neither Hindi nor English newspapers
(b) If she reads Hindi newspaper, find the probability that she reads English newspaper.
Answer:
H :
E :
The probability that she reads English newspapers if she reads Hindi newspapers
(c) If she reads English newspaper, find the probability that she reads Hindi newspaper.
Answer:
H :
E :
The probability that she reads a Hindi newspaper if she reads an English newspaper
Question 17: The probability of obtaining an even prime number on each die, when a pair of dice is rolled is
Answer:
When a pair of dice is rolled, total outcomes
Even prime number
The probability of obtaining an even prime number on each die
Option D is correct.
Question 18: Two events A and B will be independent if
(A)
Answer:
Two events A and B will be independent if
Or
Option B is correct.
Class 12 Maths chapter 13 solutions - Exercise: 13.3 Page number: 431-433 Total questions: 14 |
Answer:
Black balls = 5
Red balls = 5
Total balls = 10
CASE 1 Let red ball be drawn in first attempt.
Now two red balls are added in urn .
Now red balls = 7, black balls = 5
Total balls = 12
CASE 2
Let black ball be drawn in first attempt.
Now two black balls are added in urn .
Now red balls = 5, black balls = 7
Total balls = 12
The probability that the second ball is red =
Answer:
BAG 1 : Red balls =4 Black balls=4 Total balls = 8
BAG 2 : Red balls = 2 Black balls = 6 Total balls = 8
B1 : selecting bag 1
B2 : selecting bag 2
Let R be an event of getting red ball
Probability that the ball is drawn from the first bag,
given that it is red is
Using Bayes' theorem, we have
Answer:
H : resides in a hostel
D : day scholars
A : students who attain grade A
By Bayes' theorem :
Answer:
A : Student knows the answer.
B : Student, guess the answer
C : Answer is correct
By Bayes' theorem :
Answer:
A : Person selected is having the disease
B : Person selected is not having the disease.
C :Blood result is positive.
By Bayes' theorem :
Answer:
Given : A : chossing a two headed coin
B : chossing a biased coin
C : chossing a unbiased coin
D : event that coin tossed shows a head.
Biased coin that comes up heads
Answer:
Let A : scooter drivers = 2000
B : car drivers = 4000
C : truck drivers = 6000
Total drivers = 12000
D : the event that a person meets with an accident.
Answer:
A : Items produced by machine A
B : Items produced by machine B
X : Produced item found to be defective.
Hence, the probability that a defective item was produced by machine
Answer:
A: the first groups will win
B: the second groups will win
X: Event of introducing a new product.
Probability of introducing a new product if the first group wins :
Probability of introducing a new product if the second group wins :
Hence, the probability that the new product introduced was by the second group :
Answer:
Let A: Outcome on die is 5 or 6.
B: Outcome on die is 1,2,3,4
X: Event of getting exactly one head.
Probability of getting exactly one head when she tosses a coin three times :
Probability of getting exactly one head when she tosses a coin one time :
Hence, the probability that she threw
Answer:
Let A: time consumed by machine A
B: time consumed by machine B
C: time consumed by machine C
Total drivers = 12000
D: Event of producing defective items
Hence, the probability that a defective item was produced by
=
Answer:
Let A : Event of choosing a diamond card.
B : Event of not choosing a diamond card.
X : The lost card.
If lost card is diamond, then 12 diamond cards are left out of 51 cards.
Two diamond cards are drawn out of 12 diamond cards in
Similarly, two cards are drawn out of 51 cards in
Probability of getting two diamond cards when one diamond is lost :
If the lost card is not diamond, then 13 diamond cards are left out of 51 cards.
Two diamond cards are drawn out of 13 diamond cards in
Similarly, two cards are drawn out of 51 cards in
Probability of getting two diamond cards when one diamond is not lost :
The probability of the lost card being a diamond :
Hence, the probability of the lost card being a diamond :
Answer:
Let A : A speaks truth
B : A speaks false
X : Event that head appears.
A coin is tossed, outcomes are head or tail.
Probability of getting a head whether A speaks thruth or not is
The probability that actually there was head is
Hence, option A is correct.
Question 14: If
Answer:
If
Also,
We know that
Hence, we can see that option C is correct.
Class 12 Maths chapter 13 solutions - Miscellaneous Exercise Page number: 435-437 Total questions: 13 |
Question 1(i): A and B are two events such that
Answer:
A and B are two events such that
Question 1(ii):
Answer:
A and B are two events such that
Question 2 (i): A couple has two children,
Answer:
A couple has two children,
sample space
Let A be both children are males and B is at least one of the children is male.
Question 2 (ii): A couple has two children,
Answer:
A couple has two children,
sample space
Let A be both children are females and B be the elder child is a female.
Answer:
We have
Percentage of people with grey hairs
The probability that the selected haired person is male :
Answer:
at most
the probability that more than
The probability that at most 6 of a random sample of 10 people are right-handed is given by
Question 5: If a leap year is selected at random, what is the chance that it will contain 53 Tuesdays?
Answer:
In a leap year, there are 366 days.
In 52 weeks, there are 52 Tuesdays.
The probability that a leap year will have 53 Tuesday is equal to the probability that the remaining 2 days are Tuesdays.
The remaining 2 days can be :
1. Monday and Tuesday
2. Tuesday and Wednesday
3. Wednesday and Thursday
4. Thursday and Friday
5. Friday and Saturday
6. Saturday and Sunday
7. Sunday and Monday
Total cases = 7.
Favorable cases = 2
Probability of having 53 Tuesdays in a leap year = P.
Question 6: Suppose we have four boxes A, B, C and D containing coloured marbles as given below:
(i) from box A?
(ii) from box B?
(iii) from box C?
Answer:
'
(i)
Let R be the event of drawing a red marble.
Let
Total marbles = 40
Red marbles =15
Probability of drawing red marble from box A is
(ii)
Let R be the event of drawing a red marble.
Let
Total marbles = 40
Red marbles =15
Probability of drawing red marble from box B is
(iii)
Let R be the event of drawing a red marble.
Let
Total marbles = 40
Red marbles =15
Probability of drawing red marble from box C is
Answer:
Let A, E1, and E2 respectively denote the event that a person has a heartbreak, the selected person followed the course of yoga and meditation, and the person adopted
the drug prescription.
The probability that the patient followed a course of meditation and yoga is
Answer:
Total number of determinants of the second order, with each element being 0 or 1 is
The values of determinant is positive in the following cases
Probability is
Evaluate the following probabilities
Answer:
Let the event in which A fails and B fails be
Evaluate the following probabilities
Answer:
Let event in which A fails and B fails be
Answer:
Let E1 and E2 respectively denote the event that red ball is transfered from bag 1 to bag 2 and a black ball is transfered from bag 1 to bag2.
Let A be the event that ball drawn is red.
When a red ball is transfered from bag 1 to bag 2.
When a black ball is transferred from bag 1 to bag 2.
Question 11: If A and B are two events such that
Choose the correct answer of the following:
Answer:
A and B are two events such that
Option A is correct.
Question 12: If
Answer:
Option C is correct.
Question 13: If A and B are any two events such that
Answer:
Option B is correct.
Also, read,
Question: Eight coins are tossed together. The probability of getting exactly 3 heads is:
Solution:
Given:
probability distribution
The total number of coins is tossed,
The probability of getting head,
The probability of getting tail,
The Required probability
Hence, the correct answer is
The topics discussed in the NCERT Solutions for class 12, chapter 13, Probability are:
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Conditional probability is the likelihood of an event occurring based on the occurrence of a preceding event. For two events A and B with the same sample space, the conditional probability of event A given that B has occurred (P(A|B)) is defined as:
P ( A ∣ B ) = P ( A ∩ B ) P ( B ) (when P(B) ≠ 0)
P ( S ∣ F ) = P ( F ∣ F ) = 1 P ( ( A ∪ B ) ∣ F ) = P ( A ∣ F ) + P ( B ∣ F ) − P ( ( A ∩ B ) ∣ F ) P ( E ′ ∣ F ) = 1 − P ( E ∣ F )
Multiplication Rule: The multiplication rule relates the probability of two events E and F in a sample space S:
P ( E ∩ F ) = P ( E ) ⋅ P ( F ∣ E ) = P ( F ) ⋅ P ( E ∣ F ) (when P(E) ≠ 0 and P(F) ≠ 0)
Independent Events: Two experiments are considered independent if the probability of the events E and F occurring simultaneously is the product of their individual probabilities:
P ( E ∩ F ) = P ( E ) ⋅ P ( F )
Bayes’ theorem deals with events E1, E2, …, En that form a partition of the sample space S. It allows the calculation of the probability of event Ei given event A:
P ( E i ∣ A ) = P ( E i ) ⋅ P ( A ∣ E i ) ∑ j = 1 n P ( E j ) ⋅ P ( A ∣ E j ) , for i = 1, 2, …, n
Given a partition E1, E2, …, En of the sample space and an event A, the theorem of total probability states:
P ( A ) = P ( E 1 ) ⋅ P ( A ∣ E 1 ) + P ( E 2 ) ⋅ P ( A ∣ E 2 ) + ⋯ + P ( E n ) ⋅ P ( A ∣ E n )
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The links below allow students to access all the Maths solutions from the NCERT book.
Also, read,
Here are the subject-wise links for the NCERT solutions of class 12:
Given below are the class-wise solutions of class 12 NCERT:
Here are some useful links for the NCERT books and the NCERT syllabus for class 12
To solve Bayes' Theorem questions, apply the formula:
P(A | B) = (P(B | A) x P(A)) / P(B)
Here, P(A | B) is the probability of event A given event B, P(B | A) is the probability of event B given A, and P(A) and P(B) are the probabilities of A and B.
The formula for Conditional Probability is:
P(A | B) = P(A ∩ B) / P(B)
where P(A | B) is the probability of A occurring given that B has occurred.
The basic concepts include understanding random experiments, sample spaces, events (simple and compound), probability rules (addition and multiplication rules), and probability distributions.
To solve these questions, use the Binomial Distribution formula:
P(X = k) = (n choose k) x p^k x (1 - p)^(n - k)
where n is the number of trials, k is the number of successes, and p is the probability of success in each trial.
Classical Probability is based on equally likely outcomes (i.e., all outcomes are equally probable), while Conditional Probability is the probability of an event occurring given that another event has already occurred. The key difference is that Classical looks at all possible outcomes equally, while Conditional focuses on a subset of outcomes based on additional information.
Changing from the CBSE board to the Odisha CHSE in Class 12 is generally difficult and often not ideal due to differences in syllabi and examination structures. Most boards, including Odisha CHSE , do not recommend switching in the final year of schooling. It is crucial to consult both CBSE and Odisha CHSE authorities for specific policies, but making such a change earlier is advisable to prevent academic complications.
Hello there! Thanks for reaching out to us at Careers360.
Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.
Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!
Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.
If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.
Let me know if you need any other tips for your math prep. Good luck with your studies!
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I hope this information helps you.
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If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.
hope this helps.
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