CBSE Class 12 Mathematics Probability Worksheet Set E

Question. Suppose that five good fuses and two defective ones have been mixed up. To find the defective fuses, we test them one-by-one, at random and without replacement. What is the probability that we are lucky and find both of the defective fuses in the first two tests?
(a) 1/42
(b) 2/21
(c) 1/18
(d) 1/21

Answer: D

Question.  If six cards are selected at random (without replacement) from a standard deck of 52 cards, then what is the probability that there will be no pairs (two cards of same denomination)?
(a) 0.28
(b) 0.562
(c) 0.345
(d) 0.832

Answer: C

Question. A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the event “number obtained is even” and B be the event “number obtained is red”. Find P(A ∩ B)
(a) 1/2
(b) 1/4
(c) 1/6
(d) 1/3

Answer: C

Question. If P(A) = 7/13, P(B) = 9/13 and P(A ∪ B) = 12/13, then evaluate P(A | B).
(a) 4/13
(b) 4/9
(c) 9/13
(d) 4/5

Answer: B

Question. If P(A) = 2/5 , P(B) = 3/10 and P(A ∩ B) = , 1/5 then P(A′ | B′) is equal to
(a) 5/6
(b) 5/7
(c) 25/42
(d) 1

Answer: B

Question.  A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement the probability of getting exactly one red ball is
(a) 45/196
(b) 135/392
(c) 15/56
(d) 15/29

Answer: C

Question. Let A and B be independent events with P(A) = 1/4 and P(A ∪ B) = 2P(B) – P(A). Find P(B).
(a) 1/4
(b) 3/5
(c) 2/3
(d) 2/5

Answer: D

Question. A random variable X has the following distribution.

For the event E = {X is prime number}, find P(E).
(a) 0.87
(b) 0.62
(c) 0.35
(d) 0.50

Answer: B

Question. If A and B are two events such that P(A|B) = p, P(A) = p, P(B) = 1/3 and P(A ∪ B) = 5/9,  then find the value of p.
(a) 2/3
(b) 4/9
(c) 5/9
(d) 1/3

Answer: D

Question. A bag contains 3 white and 6 black balls while another bag contains 6 white and 3 black balls. A bag is selected at random and a ball is drawn. Find the probability that the ball drawn is of white colour.
(a) 3/4
(b) 5/4
(c) 1/4
(d) 1/2

Answer: D

Question. Two dice are thrown together. What is the probability that the sum of the number on the two faces is neither 9 nor 11 ?
(a) 3/4
(b) 1/2
(c) 5/6
(d) 2/3

Answer: C

Question. If A and B are two events and A ≠ Φ, B ≠ Φ, then
(a) P(A|B) = P(A) . P(B)
(b) P(A|B) = P(A ∩ B)/P(B)
(c) P(A | B) . P(B | A) = 1
(d) P(A | B) = P(A) | P(B)

Answer: B

Question. If A and B are two independent events such that P(A ∪ B) = 0.6 and P(A) = 0.2, then find P(B).
(a) 0.3
(b) 0.4
(c) 0.1
(d) 0.5

Answer: D

Question. If A and B are two independent events, then the probability of occurrence of at least one of A and B is given by
(a) 1 – P(A) P(B)
(b) 1 – P(A) P(B′)
(c) 1 – P(A′) P(B′)
(d) 1 – P(A′) P(B)

Answer: C

Question. The probability distribution of a discrete random variable X is given below :

The value of k is
(a) 8
(b) 16
(c) 32
(d) 48

Answer: C

Question. An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement, then the probability that both drawn balls are black, is
(a) 2/7
(b) 1/7
(c) 5/7
(d) 3/7

Answer: D

Question. The probability that student entering a university will graduate is 0.4. Find the probability that out of 3 students of the university none will graduate.
(a) 0.216
(b) 0.36
(c) 0.6
(d) 0.1296

Answer: A

Question. If two events A and B are such that P(A) = 0.3, P(B) = 0.4 and P(A ∩ B) = 0.5 then P(B|(A ∪ B)) =
(a) 1/2
(b) 1/3
(c) 2/5
(d) 1/4

Answer: D

Question. If P(A) = 3/10, P(B) = 2/5 and P(A ∪ B) = 3/5 then P(B | A) + P(A | B) equals
(a) 1/4
(b) 1/3
(c) 5/12
(d) 7/12

Answer: D

Question. A problem in mathematics is given to 3 students whose chances of solving it are 1/2 ,1/3, 1/4 . What is the probability that the problem is solved ?
(a) 1/5
(b) 1/4
(c) 3/4
(d) 2/3

Answer: C

Question. A random variable X has the following probability distribution :

Find the value of a.
(a) 1/47
(b) 1/48
(c) 1/33
(d) 1/29

Answer: B

Question. The probability distribution of a discrete random variable X is given below :

The value of k is
(a) 8
(b) 16
(c) 32
(d) 48

Answer: C

Question. The probability distribution of X is

Then find P(X ≤ 1).
(a) 0.1
(b) 0.3
(c) 0.4
(d) 0.5

Answer: B

Question. If A and B are events such that P(A) > 0 and P(B) ≠ 1, then P(A′ | B′) equals
(a) 1 – P(A|B)
(b) 1 – P(A′|B)
(c) 1 – P(A ∪ B)/P(B′)
(d) P(A′) | P(B′)

Answer: C

Question. A flashlight has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, the probability that both are dead is
(a) 33/56
(b) 9/64
(c) 1/14
(d) 3/28

Answer: D

Question. You are given that A and B are two events such that P(B) = 3/5 ,P(A|B) = 1/2 and P(A ∪ B) = , 4/5 then P(A) equals
(a) 3/10
(b) 1/5
(c) 1/2
(d) 3/5

Answer: C

Question. A and B are events such that P(A) = 0.4, P(B) = 0.3 and P(A ∪ B) = 0.5. Then P(B′ ∩ A) equals
(a) 2/3
(b) 1/2
(c) 3/10
(d) 1/5

Answer: D

Question. A box contains 3 orange balls, 3 green balls and 2 blue balls. Three balls are drawn at random from the box without replacement. The probability of drawing 2 green balls and one blue ball is
(a) 3/28
(b) 2/21
(c) 1/28
(d) 167/168

Answer: A

Question. Two events A and B will be independent, if
(a) A and B are mutually exclusive
(b) P(A′ ∩ B′) = [1 – P(A)] [1 – P(B)]
(c) P(A) = P(B)
(d) P(A) + P(B) = 1

Answer: B

Question. Given that, the events A and B are such that P(A) = 1/2, P(A ∪ B) = 3/5 and P(B) = p. Then find the value of p, if A and B are mutually exclusive.
(a) 3/5
(b) 1/5
(c) 2/3
(d) 1/10

Answer: D

Question. If P(A) = 3/10, P(B) = 2/5 and P(A ∪ B) = 3/5, then find the value of P(B / A).
(a) 2/3
(b) 1/3
(c) 2/5
(d) 1/4

Answer: B

Question. If A and B are two events such that P(A) = 0.2 , P(B) = 0.4 and P(A ∪ B) = 0.5 , then value of P(A/B) is
(a) 0.1
(b) 0.25
(c) 0.5
(d) 0.08

Answer: B

Question. An urn contains 6 balls of which two are red and four are black. Two balls are drawn at random. Probability that they are of the different colours is
(a) 2/5
(b) 1/15
(c) 8/15
(d) 4/15

Answer: C

Question. If P(A) = 0.4, P(B) = 0.8 and P(B | A) = 0.6, then P(A ∪ B) is equal to
(a) 0.24
(b) 0.3
(c) 0.48
(d) 0.96

Answer: D

Case Based MCQs

Case I : Read the following passage and answer the questions.

A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by cab, metro, bike or by other means of transport are respectively 0.3, 0.2, 0.1 and 0.4. The probabilities that he will be late are 0.25, 0.3, 0.35 and 0.1 if he comes by cab, metro, bike and other means of transport respectively.

Question. When the doctor arrives late, what is the probability that he comes by metro?
(a) 5/14
(b) 2/7
(c) 5/21
(d) 1/6

Answer: B

Question. When the doctor arrives late, what is the probability that he comes by cab?
(a) 4/21
(b) 1/7
(c) 5/14
(d) 2/21

Answer: C

Question. When the doctor arrives late, what is the probability that he comes by bike?
(a) 5/21
(b) 4/7
(c) 56/
(d) 1/6

Answer: D

Question. When the doctor arrives late, what is the probability that he comes by other means of transport?
(a) 6/7
(b) 5/14
(c) 4/21
(d) 2/7

Answer: C

Question. What is the probability that the doctor is late by any means?
(a) 1
(b) 0
(c) 1/2
(d) 1/4

Answer: A

Case II : Read the following passage and answer the questions.

One day, a sangeet mahotsav is to be organised in an open area of Rajasthan. In recent years, it has rained only 6 days each year. Also, it is given that when it actually rains, the weatherman correctly forecasts rain 80% of the time. When it doesn’t rain, he incorrectly forecasts rain 20% of the time. If leap year is considered, then answer the following questions.

Question. The probability that it rains on chosen day is
(a) 1/366
(b) 1/73
(c) 1/60
(d) 1/61

Answer: B

Question. The probability that it does not rain on chosen day is
(a) 1/366
(b) 5/366
(c) 360/366
(d) none of these

Answer: C

Question. The probability that the weatherman predicts correctly is
(a) 56
(b) 78
(c) 45
(d) 15

Answer: C

Question. The probability that it will rain on the chosen day, if weatherman predict rain for that day, is
(a) 0.0625
(b) 0.0725
(c) 0.0825
(d) 0.0925

Answer: A

Question. The probability that it will not rain on the chosen day, if weatherman predict rain for that day, is
(a) 0.94
(b) 0.84
(c) 0.74
(d) 0.64

Answer: A

Case III : Read the following passage and answer the questions.

Varun and Isha decided to play with dice to keep themselves busy at home as their schools are closed due to coronavirus pandemic. Varun throw a dice repeatedly until a six is obtained. He denote the number of throws required by X.

Question. The probability that X = 2 equals
(a) 1/6
(b) 5/6²
(c) 5/3⁶
(d) 1/6³

Answer: B

Question. The probability that X = 4 equals
(a) 1/6⁴
(b) 1/6⁶
(c) 5³/6⁴
(d) 5/6⁴

Answer: C

Question. The probability that X ≥ 2 equals
(a) 25/216
(b) 1/36
(c) 56/
(d) 25/36

Answer: C

Question. The value of P(X ≥ 6) is
(a) 5⁵/6⁵
(b) 1 – 5³/6⁵
(c) 5³ x 61 / 6⁵
(d) 5³/6⁴

Answer: A

Question. The probability that X > 3 equals
(a) 36'/25
(b) 5²/6²
(c) 5/6
(d) 5³/6³

Answer: D

Assertion & Reasoning Based MCQs

Directions : In these questions, a statement of Assertion is followed by a statement of Reason is given. Choose the correct answer out of the following choices :
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.

Question. Let E₁ and E₂ be any two events associated with an experiment, then
Assertion : P(E₁) + P(E₂) ≤ 1.
Reason : P(E₁) + P(E₂) = P(E₁ ∪ E₂) + P(E₁ ∩ E₂).

Answer: D

52. Consider the system of equations ax + by = 0, cx + dy = 0 where a, b, c, d ∈{0, 1}.
Assertion : The probability that the system of equations has a unique solution is 3/8
Reason : The probability that the system of equations has a solution is 1.

Answer: B

53. Assertion : Consider the experiment of  drawing a card from a deck of 52 playing cards, in which the elementary events are assumed to be equally likely If E and F denote the events the card drawn is a spade and the card drawn is an ace respectively, then P(E|F) = 1'/4 and P(F|E) = . 1/13
Reason : E and F are two events such that the probability of occurrence of one of them is not dffected by occurrence of the other. Such events are called independent events.

Answer: A

54. Assertion : Let E and F be events associated with the sample space S of an experiment. Then, we have P(S|F) = P(F|F) = 1.
Reason : If A and B are any two events associated with the sample space S and F is an event associated with S such that P(F) ≠ 0, then P((A ∪ B)|F) = P(A|F) + P(B|F) – P((A ∩ B)|F)

Answer: B

Question. Let A and B be two events associated with an experiment such that P(A ∩ B) = P(A)P(B).
Assertion : P(A|B) = P(A) and P(B|A) = P(B) 
Reason : P(A ∪ B) = P(A) + P(B).

Answer: C

Question. Assertion : Bag I contains 3 red and 4 black balls while another bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Then, the probability that it was drawn from bag II is 35/68.
Reason : 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, then the probability that the other coin in the box is also of gold is 1/2

Answer: C

Question. Assertion : 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. Then, the probability that the second ball is red is 1/2
Reason : A bag contains 4 red and 4 black balls, 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. Then, the probability that the ball is drawn form the first bag is 2/3.

Answer: B

Question. Assertion : The probability that candidates A and B can solve the problem is 1/5 and 2/5, then probability that problem will be solved is given by 12/25 .
Reason : If events A & B are independent, then P (A ∩ B) = P (A) × P (B).

Answer: D

Question. A man P speaks truth with probability p and an other man Q speaks truth with probability 2p.
Assertion : If P and Q contradict each other with probability 1/2, then there are two values of p.
Reason : A quadratic equation with real coefficients has two real roots.

Answer: C