CBSE Class 12 Mathematics Probability MCQs Set 09

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) \( \frac{2}{7} \)
(b) \( \frac{1}{7} \)
(c) \( \frac{5}{7} \)
(d) \( \frac{3}{7} \)
Answer: (d) \( \frac{3}{7} \)

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) 0.216

Question. If two events A and B are such that \( P(A') = 0.3 \), \( P(B) = 0.4 \) and \( P(A \cap B') = 0.5 \), then \( P(B | (A \cup B')) \)
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{2}{5} \)
(d) \( \frac{1}{4} \)
Answer: (d) \( \frac{1}{4} \)

Question. Let X denote the number of hours you study on a Sunday. Also it is known that
\( P(X = x) = \begin{cases} 0.2 & \text{if } x = 0 \\ kx & \text{if } x = 1 \text{ or } 2 \\ k(4-x) & \text{if } x = 3 \text{ or } 4 \\ 0 & \text{otherwise} \end{cases} \)
where k is a constant. What is the probability that you study atleast two hours?
(a) 0.55
(b) 0.15
(c) 0.6
(d) 0.3
Answer: (c) 0.6

Question. If \( P(A) = \frac{3}{10} \), \( P(B) = \frac{2}{5} \) and \( P(A \cup B) = \frac{3}{5} \), then \( P(B | A) + P(A | B) \) equals
(a) \( \frac{1}{4} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{5}{12} \)
(d) \( \frac{7}{12} \)
Answer: (d) \( \frac{7}{12} \)

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

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

Find the value of a.
(a) \( \frac{1}{47} \)
(b) \( \frac{1}{48} \)
(c) \( \frac{1}{33} \)
(d) \( \frac{1}{29} \)
Answer: (b) \( \frac{1}{48} \)

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

Question. The probability distribution of X is:

Then find \( P(X \leq 1) \).
(a) 0.1
(b) 0.3
(c) 0.4
(d) 0.5
Answer: (b) 0.3

Question. If A and B are events such that \( P(A) > 0 \) and \( P(B) \neq 1 \), then \( P(A' | B') \) equals
(a) \( 1 - P(A|B) \)
(b) \( 1 - P(A'|B) \)
(c) \( \frac{1 - P(A \cup B)}{P(B')} \)
(d) \( P(A') | P(B') \)
Answer: (c) \( \frac{1 - P(A \cup B)}{P(B')} \)

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) \( \frac{33}{56} \)
(b) \( \frac{9}{64} \)
(c) \( \frac{1}{14} \)
(d) \( \frac{3}{28} \)
Answer: (d) \( \frac{3}{28} \)

Question. You are given that A and B are two events such that \( P(B) = \frac{3}{5} \), \( P(A|B) = \frac{1}{2} \) and \( P(A \cup B) = \frac{4}{5} \), then P(A) equals
(a) \( \frac{3}{10} \)
(b) \( \frac{1}{5} \)
(c) \( \frac{1}{2} \)
(d) \( \frac{3}{5} \)
Answer: (c) \( \frac{1}{2} \)

Question. A and B are events such that \( P(A) = 0.4 \), \( P(B) = 0.3 \) and \( P(A \cup B) = 0.5 \). Then \( P(B' \cap A) \) equals
(a) \( \frac{2}{3} \)
(b) \( \frac{1}{2} \)
(c) \( \frac{3}{10} \)
(d) \( \frac{1}{5} \)
Answer: (d) \( \frac{1}{5} \)

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) \( \frac{3}{28} \)
(b) \( \frac{2}{21} \)
(c) \( \frac{1}{28} \)
(d) \( \frac{167}{168} \)
Answer: (a) \( \frac{3}{28} \)

Question. Two events A and B will be independent, if
(a) A and B are mutually exclusive
(b) \( P(A' \cap B') = [1 - P(A)][1 - P(B)] \)
(c) \( P(A) = P(B) \)
(d) \( P(A) + P(B) = 1 \)
Answer: (b) \( P(A' \cap B') = [1 - P(A)][1 - P(B)] \)

Question. Given that, the events A and B are such that \( P(A) = \frac{1}{2} \), \( P(A \cup B) = \frac{3}{5} \) and \( P(B) = p \). Then find the value of p, if A and B are mutually exclusive.
(a) \( \frac{3}{5} \)
(b) \( \frac{1}{5} \)
(c) \( \frac{2}{3} \)
(d) \( \frac{1}{10} \)
Answer: (d) \( \frac{1}{10} \)

Question. If \( P(A) = \frac{3}{10} \), \( P(B) = \frac{2}{5} \) and \( P(A \cup B) = \frac{3}{5} \), then find the value of \( P(B / A) \).
(a) \( \frac{2}{3} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{2}{5} \)
(d) \( \frac{1}{4} \)
Answer: (b) \( \frac{1}{3} \)

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

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) \( \frac{2}{5} \)
(b) \( \frac{1}{15} \)
(c) \( \frac{8}{15} \)
(d) \( \frac{4}{15} \)
Answer: (c) \( \frac{8}{15} \)

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

Case Based MCQs

Case I: Read the following passage and answer the questions from 46 to 50.
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) \( \frac{1}{6} \)
(b) \( \frac{5}{6^2} \)
(c) \( \frac{5}{36} \)
(d) \( \frac{1}{6^3} \)
Answer: (b) \( \frac{5}{6^2} \)

Question. The probability that X = 4 equals
(a) \( \frac{1}{6^4} \)
(b) \( \frac{1}{6^6} \)
(c) \( \frac{5^3}{6^4} \)
(d) \( \frac{5}{6^4} \)
Answer: (c) \( \frac{5^3}{6^4} \)

Question. The probability that X ≥ 2 equals
(a) \( \frac{25}{216} \)
(b) \( \frac{1}{36} \)
(c) \( \frac{5}{6} \)
(d) \( \frac{25}{36} \)
Answer: (c) \( \frac{5}{6} \)

Question. The value of P(X ≥ 6) is
(a) \( \left(\frac{5}{6}\right)^5 \)
(b) \( 1 - \left(\frac{5}{6}\right)^5 \)
(c) \( \frac{5^3 \times 61}{6^5} \)
(d) \( \left(\frac{5}{6}\right)^4 \)
Answer: (a) \( \left(\frac{5}{6}\right)^5 \)

Question. The probability that X > 3 equals
(a) \( \frac{36}{25} \)
(b) \( \left(\frac{5}{6}\right)^2 \)
(c) \( \frac{5}{6} \)
(d) \( \left(\frac{5}{6}\right)^3 \)
Answer: (d) \( \left(\frac{5}{6}\right)^3 \)

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 \( H_1, H_2, \dots, H_n \) be mutually exclusive and exhaustive events with \( P(H_i) > 0, i = 1, 2, \dots, n \). Let E be any other event with \( 0 < P(E) < 1 \).
Assertion : \( P(H_i | E) > P(E | H_i)P(H_i) \) for \( i = 1, 2, \dots, n \).
Reason : \( \sum_{i=1}^{n} P(H_i) = 1 \).
(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: (d) Assertion is wrong statement but Reason is correct statement.

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 \( \frac{35}{68} \).
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 \( \frac{1}{2} \).
(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: (c) Assertion is correct statement but Reason is wrong statement.

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 \( \frac{1}{2} \).
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 from the first bag is \( \frac{2}{3} \).
(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. Assertion : The probability that candidates A and B can solve the problem is \( \frac{1}{5} \) and \( \frac{2}{5} \), then probability that problem will be solved is given by \( \frac{12}{25} \).
Reason : If events A & B are independent, then \( P(A \cap B) = P(A) \times P(B) \).
(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: (d) Assertion is wrong statement but Reason is correct statement.

Question. A man P speaks truth with probability p and another 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.
(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.