CBSE Class 12 Mathematics Probability MCQs Set 08

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) \( \frac{1}{42} \)
(b) \( \frac{2}{21} \)
(c) \( \frac{1}{18} \)
(d) \( \frac{1}{21} \)
Answer: (d) \( \frac{1}{21} \)

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) 0.345

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 \( P(A \cap B) \)
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{4} \)
(c) \( \frac{1}{6} \)
(d) \( \frac{1}{3} \)
Answer: (c) \( \frac{1}{6} \)

Question. If \( P(A) = \frac{7}{13} \), \( P(B) = \frac{9}{13} \) and \( P(A \cup B) = \frac{12}{13} \), then evaluate \( P(A | B) \).
(a) \( \frac{4}{13} \)
(b) \( \frac{4}{9} \)
(c) \( \frac{9}{13} \)
(d) \( \frac{4}{5} \)
Answer: (b) \( \frac{4}{9} \)

Question. If \( P(A) = \frac{2}{5} \), \( P(B) = \frac{3}{10} \) and \( P(A \cap B) = \frac{1}{5} \), then \( P(A' | B') \) is equal to
(a) 5/6
(b) 5/7
(c) 25/42
(d) 1
Answer: (b) 5/7

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) \( \frac{45}{196} \)
(b) \( \frac{135}{392} \)
(c) \( \frac{15}{56} \)
(d) \( \frac{15}{29} \)
Answer: (c) \( \frac{15}{56} \)

Question. Let A and B be independent events with \( P(A) = 1/4 \) and \( P(A \cup B) = 2P(B) - P(A) \). Find \( P(B) \).
(a) \( \frac{1}{4} \)
(b) \( \frac{3}{5} \)
(c) \( \frac{2}{3} \)
(d) \( \frac{2}{5} \)
Answer: (d) \( \frac{2}{5} \)

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) 0.62

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

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) \( \frac{3}{4} \)
(b) \( \frac{5}{4} \)
(c) \( \frac{1}{4} \)
(d) \( \frac{1}{2} \)
Answer: (d) \( \frac{1}{2} \)

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) \( \frac{3}{4} \)
(b) \( \frac{1}{2} \)
(c) \( \frac{5}{6} \)
(d) \( \frac{2}{3} \)
Answer: (c) \( \frac{5}{6} \)

Question. If A and B are two events and \( A \neq \phi, B \neq \phi \), then
(a) \( P(A|B) = P(A) \cdot P(B) \)
(b) \( P(A|B) = \frac{P(A \cap B)}{P(B)} \)
(c) \( P(A | B) \cdot P(B | A) = 1 \)
(d) \( P(A | B) = P(A) | P(B) \)
Answer: (b) \( P(A|B) = \frac{P(A \cap B)}{P(B)} \)

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

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) \( 1 - P(A') P(B') \)

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) 32

Case Based MCQs

Case I: Read the following passage and answer the questions from 36 to 40.
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) \( \frac{5}{14} \)
(b) \( \frac{2}{7} \)
(c) \( \frac{5}{21} \)
(d) \( \frac{1}{6} \)
Answer: (b) \( \frac{2}{7} \)

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

Question. When the doctor arrives late, what is the probability that he comes by bike?
(a) \( \frac{5}{21} \)
(b) \( \frac{4}{7} \)
(c) \( \frac{5}{6} \)
(d) \( \frac{1}{6} \)
Answer: (d) \( \frac{1}{6} \)

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

Question. What is the probability that the doctor is late by any means?
(a) 1
(b) 0
(c) \( \frac{1}{2} \)
(d) \( \frac{1}{4} \)
Answer: (a) 1

Case II: Read the following passage and answer the questions from 41 to 45.
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) \( \frac{1}{366} \)
(b) \( \frac{1}{73} \)
(c) \( \frac{1}{60} \)
(d) \( \frac{1}{61} \)
Answer: (d) \( \frac{1}{61} \)

Question. The probability that it does not rain on chosen day is
(a) \( \frac{1}{366} \)
(b) \( \frac{5}{366} \)
(c) \( \frac{360}{366} \)
(d) None of the options
Answer: (c) \( \frac{360}{366} \)

Question. The probability that the weatherman predicts correctly is
(a) \( \frac{5}{6} \)
(b) \( \frac{7}{8} \)
(c) \( \frac{4}{5} \)
(d) \( \frac{1}{5} \)
Answer: (c) \( \frac{4}{5} \)

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) 0.0625

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) 0.94

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 \( E_1 \) and \( E_2 \) be any two events associated with an experiment, then
Assertion : \( P(E_1) + P(E_2) \leq 1 \).
Reason : \( P(E_1) + P(E_2) = P(E_1 \cup E_2) + P(E_1 \cap E_2) \).
(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.
Answer: (d) Assertion is wrong statement but Reason is correct statement.

Question. Consider the system of equations \( ax + by = 0, cx + dy = 0 \) where \( a, b, c, d \in \{0, 1\} \).
Assertion : The probability that the system of equations has a unique solution is \( \frac{3}{8} \).
Reason : The probability that the system of equations has a solution is 1.
(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.
Answer: (b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Question. 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.
Assertion : If E and F denote the events the card drawn is a spade and the card drawn is an ace respectively, then \( P(E|F) = \frac{1}{4} \) and \( P(F|E) = \frac{1}{13} \).
Reason : E and F are two events such that the probability of occurrence of one of them is not affected by occurrence of the other. Such events are called independent events.
(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.
Answer: (a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Question. 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 \( P(F) \neq 0 \), then \( P((A \cup B)|F) = P(A|F) + P(B|F) - P((A \cap B)|F) \).
(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.
Answer: (b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Question. Let A and B be two events associated with an experiment such that \( P(A \cap B) = P(A)P(B) \).
Assertion : \( P(A|B) = P(A) \) and \( P(B|A) = P(B) \).
Reason : \( P(A \cup B) = P(A) + P(B) \).
(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.
Answer: (c) Assertion is correct statement but Reason is wrong statement.