Important Questions for CBSE Class 12 Maths Chapter 13 - Probability

CBSE Class 12 Maths Chapter-13 Important Questions – Free PDF Download

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CBSE Class 12 Mathematics Important Questions Chapter 13 – Probability

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4 Mark Questions

1. In a school there are 1000 students, out of which 430 are girls. It is known that out of 430, 10% of the girls study in class XII. What is the probability that a student chosen randomly studies in class XII given that the chosen student is a girl?
Ans. Let E, student chosen randomly studies in class XII, F randomly chosen student is girl.
P (E|F) = ?



= = 0.1

2. A die thrown three times. Events A and B are defined as below.
A : 4 on the third throw
B : 6 on the first and 5 on the second throw.
Find the probability of A given that B has already occurred.
Ans. Total sample space = 216

3.Mother, father and son line up at random for a family picture
E : Son on one end
F : Father in middle
Find (E |F)
Ans.

4.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, what is the probability that it will be an easy question given that it is a multiple choice question?
Ans. let E : easy question
F : multiple choice question
total questions = 300 + 200 + 500 + 400
= 1400

5.If A and B are two independent events, then the probability of occurrence of at least one of A and B is given by 1 – P () P ().
Ans. P (at least one of A and B) = P (A∪B)
= P (A) + P (B) – P (A∩B)
= P (A) + P (B) – P (A). P (B)
= P (A) + P (B) [1 – P (A)]
= P (A) + P (B) P ()
= 1 – P () + P (B) P ()
= 1 – P () [1 – P (B)]
= 1 – P () P (())

6.A – Box of oranges is inspected by examining three randomly selected oranges drown without replacement. It all the three oranges are good, the box is approved for sale, other wise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approve for sale.
Ans. Required

7.A fair coin and an unbiased die are tossed. Let A be the event head appear on the coin and B be the event 3 on the die.
Check weather A and B are independent events or not.
Ans. 
A : Head appear on the coin
B : 3 appear on the lice






Hence A and B are independent

8.Probability of solving specific problem independently by A and B are and respectively of both try to solve the problem independently, find the probability that
(i) the problem is solved
(ii) Exactly one of them solves the problem.
Ans. E₁ : A solves the problem
E₂ : B solves the problem
P (E₁) = and P (E₂) = 
(i) P (the problem is solved)
= 1 – P (the problem is not solved)


(ii) P Exactly one of their solves the problem

9.In a hostel 60% of the students read Hindi news paper, 40% read English news paper and 20% read both Hindi and English news papers. A student is selected et random.
(a) Find the probability that she read neither Hindi nor English news papers.
(b) If the reads Hindi news paper, find the probability that she reads English news paper.
(c) If she reads English news papers, find the probability that she reads Hindi news paper.
Ans. E : Student read Hindi newspaper
F : Student read English newspaper

10.Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and the third card drawn is ace.
Ans. By multiplication theorem

11. Find the probability distribution of number of doublets in three throws of a pair of dice.
Ans. Let x denote the number of doublets x = 0, 1, 2, 3
Probability of getting doublet = 
Probability of not getting doublet = 

X	0	1	2	3
P (x)

12.Let X denote the no of hours you study during a randomly selectee school day. The probability that X can take the values x, has the following form where K is some unknown constant

(a) Find the value of K
(b) What is the probability that you study at least two hours.
Exactly two hours? At most 2 hr.
Ans. The probability distribution of x is

(a) 

(b) p (study atleast two hr) = p (x ≥ 2)




p (Study exactly two hr) 

p (Study et most two hr) 

13. Find the variance of the number obtained on a throw of an unbiased die.
Ans. S = {1, 2, 3, 4, 5, 6}
p (1) = p (2) = p (3) = p (4) = p (5) =p (6)
= 

x	1	2	3	4	5	6
P (x)

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6 Marks Questions

1. Given three identical boxes I, II and III each containing two coins. In box-I both coins are gold coins, in box-II, both are silver coins and in the box-III, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold.
Ans. let E₁, E₂ and E₃ be the events that boxes I, II and III are chosen.
P (E₁) = P (E₂) = P (E₃) = 
let A be the event the coin drawn is of gold.

2. Suppose that the reliability of a HIV test is specified as follows of people having HIV, 90% of the test detect the disease but 10% go undetected of people free of HIV, 99% of the test are Judged HIV – tive but 1% are diagnosed as showing HIV +tive. From a large population of which only 0.1% have HIV, one person is selected at random, given the HIV test, and the pathologist reports him/her is HIV +tive what is the probability that the person actually has HIV.
Ans. let E denote the event that the person selected is actually having HIV and A the event that the person’s HIV test is diagnosed as + tive.
let not having HIV.

3. In a factory which manufactures bolts, machines. A, B and C manufacture respectively 25%, 35% and 40% of the bolts. Of their output 5,4 and 2 percent are respectively defective bolts. A bolt is drown at random from the product and is found to be defective. What is the probability that it is manufactured by the machine B.
Ans. let B₁ = bolt is manufactures by A
B₂ = bolt is manufactured by B
B₃ = bolt is manufactured by C
let E bolt is defective
P (B₁) = 25% = 0.25
P (B₂) = 0.35
P (B₃) = 0.40
P ( E|B₁) = 5% = 0.05
P (E|B₂) = 0.04
P (E|B₃) = 0.02

4. A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by train, bus, scooter or by other mean of transport are respectively The probabilities that he will be late are if he comes by train, bus and scooter respectively, but he comes by other means of transport, that he will
 
not the late. When he arrives he is late. What is the probability that he comes by train.
Ans. let E be the event that the doctor visits the patient late and let T₁, T₂, T₃, T₄, be the event that the doctor comes by train, bus, scooter and other means of Transport respectively. 

5. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.
Ans. let E be the event that the man reports that six occurs in the throwing of the dice and let S₁ be the event that six occurs and S₂ be the event six does not occur.

6. In answering a question on a multiple choice test a student either knows the answer or guesses Let be the probability that he knows the answer and ¼ be the probability he guesses. Assuming that a student who guesses at the answer will be correct with probability . What is the probability that the student knows the answer given that he answered it correctly.
Ans. E₁ : the student knows the answer
E₂ : the student guesses the answer
A the answer is correct
P (A/E₁) = 1, P (A/E₂) = 

7. 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 test then with probability 0.005 the test will imply he has the disease) If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive.
Ans. E₁ : the person has the disease
E₂ : the person is healthy
A : test is positive
P (E₁) = 
P (A/E₁) = = 

= 
= 

8. An insurance company insured 2000 scooter drivers, 4000, car drivers and 6000 truck drivers. The probability of accidents is 0.01, 0.03 and 0.15 respectively. One of the insured persons meet with an accident what is the probability that he is scooter driver.
Ans. E₁ : Insured person is a scooter driver
E₂ : Insured person is a car driver
E₃ : Insured person is a truck driver

9. A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.
Ans. E1 : lost card is diamond
E2 : lost card is not diamond
let A : two cards drawn from the remaining pack are diamonds.




= 

10. Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3, 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one had, what is the probability that she threw 1, 2, 3 or 4 with the die?
Ans. E₁ : 1, 2, 3, 4 is shown on dice
E₂ : 5 or 6 is shown on dice
P (E₁) = 
let A exactly one head shown up

11. Two cards are drawn simultaneously (or successively without replacement) for a well shuffled of 52 cards. Find the mean, variance and standard deviation of the number of kings.
Ans. Let x denote the number of kings in a draw of two cards.

x	0	1	2
p (x)

x = E (x) = 
=

12. From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.
Ans. S = 30
A = 6 defective bulbs 

x	0	1	2	3	4
p (x

13. In a meeting 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take x = 0 if he opposed and x = 1 if he is in favour. Find E (x) and var (x).
Ans.

x	0	1
p (x)

14. A and B throw a die alternatively till one of them gets a ‘6’ and win the game. Find their respective probabilities of winning if A starts first.
Ans. S denote the success
F denote the failure

15. Find the mean of the Binomial distribution B 
Ans. Let x be the random variable whose probability distribution is B 

16. If a leap year is selected at random, what is the chance that it will contain 53 Tuesdays?
Ans. A leap contain 52 weeks and two additional days which can be Sunday and Monday, Monday and Tuesday, Tuesday and Wednesday, Wednesday and Thursday, Thursday and Friday, Friday and Saturday, Saturday and Sunday.

17. Bag I contain 3 red and 4 black balls and bag II contain 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is fund to be red in colour. Find the probability that the transferee ball is black.
Ans. Let E₁: red ball is transferred from bag I to bag II.
E₂ : black is transferred from bag I to bag II.