# MCQ Questions for Class 12 Maths Chapter 13 Probability with Answers

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## Probability Class 12 MCQs Questions with Answers

Don’t forget to practice the multitude of MCQ Questions on Probability Class 12 with answers so you can apply your skills during the exam.

Question 1.
If P(A) = $$\frac { 1 }{2}$$, P(B) = 0, thenP (A/B) is
(a) 0
(b) $$\frac { 1 }{2}$$
(c) not defined
(d) 1.

Question 2.
If A and B are events such that P (A/B) = P(B/A), then
(a) A ⊂ B but A ≠ B
(b) A = B
(c) A ∩ B = ø
(d) P (A) = P (B).

Answer: (d) P (A) = P (B).

Question 3.
The probability of obtaining an even prime number on each die when a pair of dice is rolled is
(a) 0
(b) $$\frac { 1 }{3}$$
(c) $$\frac { 1 }{12}$$
(d) $$\frac { 1 }{36}$$

Answer: (d) $$\frac { 1 }{36}$$

Question 4.
Two events A and B are said to 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) P (A’B’) = [1 – P(A)] [1 – P(B)]

Question 5.
Probability that A speaks truth is $$\frac { 4 }{5}$$. A coin is tossed. A reports that a head appears. The probability that actually there was head is:
(a) $$\frac { 4 }{5}$$
(b) $$\frac { 1 }{2}$$
(c) $$\frac { 1 }{5}$$
(d) $$\frac { 2 }{5}$$

Answer: (a) $$\frac { 4 }{5}$$

Question 6.
If A and B are two events such that A ⊂ B and P (B) ≠ 0, then which of the following is correct
(a) P (A / B) = $$\frac { p(B) }{p(A)}$$
(b) P (A/B) < P (A)
(c) P (A/B) ≥ P (A)
(d) None of these.

Answer: (c) P (A/B) ≥ P (A)

Question 7.
If A and B are two events such that P (A) ≠ 0 and P (B/A) = 1, then
(a) A ⊂ B
(b) B ⊂ A
(c) B = ø
(d) A = ø

Question 8.
If P (A/B) > P (A), then which of the following is correct?
(a) P (B/A) < P (B)
(b) P (A ∩ B) < P (A).P(B)
(c) P (B/A) > P (B)
(d) P (B/A) = P (B).

Answer: (c) P (B/A) > P (B)

Question 9.
If A and B are any two events such that
P (A) + P (B) – P (A and B) = P (A), then:
(a) P (B/A) = 1
(b) P (A/B) = 1
(c) P (B/A) = 0
(d) P (A/B) = 0

Answer: (b) P (A/B) = 1

Question 10.
Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. What is the value of E (X)?
(a) $$\frac { 37 }{221}$$
(b) $$\frac { 5 }{13}$$
(c) $$\frac { 1 }{13}$$
(d) $$\frac { 2 }{13}$$

Answer: (d) $$\frac { 2 }{13}$$

Question 11.
A die is thrown once, then the probability of getting a number greater than 3 is :
(a) $$\frac { 1 }{2}$$
(b) $$\frac { 2 }{3}$$
(c) 6
(d) 0.

Answer: (a) $$\frac { 1 }{2}$$

Question 12.
Let A and B be two events. If P(A) = 0.2, P(B) = 0.4, P(A ∪ B) = 0.6, then P(A/B) is equal to:
(a) 0.8
(b) 0.5
(c) 0.3
(d) 0.

Question 13.
Let A and B be two events such that P(A) = 0.6, P(B) = 0.2 and P(A/B) = 0.5. Then P(A’/B’) equals
(a) $$\frac { 1 }{10}$$
(b) $$\frac { 3 }{10}$$
(c) $$\frac { 3 }{8}$$
(d) $$\frac { 6 }{7}$$

Answer: (c) $$\frac { 3 }{8}$$

Question 14.
If A and B are independent events such that 0 < P(A) < 1 and 0 < P(B) < 1, then which of the following is not correct?
(a) A and B are mutually exclusive
(b) A and B’ are independent
(c) A’ and B are independent
(d) A’ and B’ are independent.

Answer: (a) A and B are mutually exclusive

Question 15.
Let ‘X’ be a discrete random variable. The probability distribution of X is given below Then E(X) is equal to
(a) 6
(b) 4
(c) 3
(d) (-5).

Question 16.
Let ‘X’ be a discrete random variable assuming values x1, x2, …………… , xn with probabilities p1, p2, …………. , pn respectively. Then variance of ‘X’ is given by
(a) E(X²)
(b) E(X²) + E(X)
(c) E(X²) – [E(X)]²
(d) $$\sqrt { E(X)^2-[E(X)]^2}$$

Question 17.
If it is given that the events A and B are such that P (A) = $$\frac { 1 }{4}$$, P (A/B) = $$\frac { 1 }{2}$$ and P(B/A) = $$\frac { 2 }{3}$$. Then P (B) is:
(a) $$\frac { 1 }{2}$$
(b) $$\frac { 1 }{6}$$
(c) $$\frac { 1 }{3}$$
(d) $$\frac { 2 }{3}$$

Answer: (c) $$\frac { 1 }{3}$$
Hint:
By definition, P (A/B) = $$\frac { P(A∩B) }{P(B)}$$
⇒ P(B) P(A/B) = P(A∩B) ……….(1)
Similarly P(A) P(B/A) = P(B∩A) ……….(2)
From (1) and (2), P(A) P(B/A)
= P(B) P(A/B)
[∵ P(A∩B) = PP(B∩A)]
⇒ $$\frac { 1 }{4}$$.$$\frac { 2}{3}$$ = P(B).$$\frac { 1 }{2}$$
⇒ P(B) = $$\frac { 1 }{4}$$.$$\frac { 2 }{3}$$.2 = $$\frac { 1 }{3}$$

Question 18.
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.

Hint:
P(A∪B) = P(A) + P(B) – P(A∩B)
⇒ 0.5 = 0.2 + 0.4 – P(A∩B)
⇒ P(A∩B) = 0.6 – 0.5 = 0.1.
∴ P(A∩B) = $$\frac { P(A∩B) }{P(B)}$$ = $$\frac { 0.1 }{0.4}$$ = 0.25

Question 19.
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) $$\frac { 8 }{15}$$
Hint:
Reqd. probability = P(RB) + P (BR)
(R ≡ Red ball and B ≡ Black ball)
= ($$\frac { 2 }{6}$$ × $$\frac { 4 }{5}$$) + ($$\frac { 4 }{6}$$ × $$\frac { 2 }{5}$$) = $$\frac { 4 }{15}$$ + $$\frac { 4 }{15}$$ = $$\frac { 8 }{15}$$

Question 20
Let A, B, C be pairwise independent events with P (C) > 0 and P (A∩B∩C) = 0. Then P (Ac∩Bc /C) is
(a) P (A) – P (Bc)
(b) P (Ac) + P (Bc)
(c) P (Ac) – P (Bc)
(d) P (Ac) – P (B).

Answer: (d) P (Ac) – P (B).
Hint: Question 21.
Three numbers are chosen at random with-out replacement from {1, 2, 3, ….. 8}. The probability that their minimum is 3, given that their maximum is 6 is
(a) $$\frac { 3 }{8}$$
(b) $$\frac { 1 }{5}$$
(c) $$\frac { 1 }{4}$$
(d) $$\frac { 2 }{5}$$

Answer: (b) $$\frac { 1 }{5}$$
Let the events be ({1, 2, 3, ….. , 8})
A : Maximum of three numbers is 6
B : Minimum of three numbers is 3. Question 22.
Let A and B be two events such that P$$(\overline{\mathbf{A} \cup \mathbf{B}})$$ = $$\frac { 1 }{6}$$ P(A∩B) = $$\frac { 1 }{4}$$ and P($$\bar { A}$$) = $$\frac { 1 }{4}$$, where $$\bar { A}$$stands for the complement of the event A. Then the events A and B are
(a) equally likely but not independent
(b) independent but not equally likely
(c) independent and equally likely
(d) mutually exclusive and independent.

Answer: (b) independent but not equally likely
P$$(\overline{\mathbf{A} \cup \mathbf{B}})$$ = $$\frac { 1 }{6}$$, P(A∪B) = $$\frac { 5 }{6}$$, P(A) = $$\frac { 3 }{4}$$.
Now P(A∪B) = P(A) + P(B) – P(A∩B) Question 23.
If two different numbers are taken from set (0, 1, 2, …… , 10}, then the probability that their sum as well as absolute difference are both multiples of 4, is
(a) $$\frac { 14 }{45}$$
(b) $$\frac { 7 }{55}$$
(c) $$\frac { 6 }{55}$$
(d) $$\frac { 12 }{55}$$

Answer: (c) $$\frac { 6 }{55}$$
Hint:
Let A = {0, 1, 2, 3, …….. , 10}
∴ n(S) = 11C2 = 55.
Let E be the given event.
∴ E = {(0, 4), (0, 8), (2, 6), (2, 10), (4, 8), (6, 10) }.
∴n(E) = 6
Hence P(E) = $$\frac { n(E) }{n(S)}$$ = $$\frac { 6 }{55}$$

Question 24.
A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag.
If now a ball is drawn at random from the bag, then the probability that this drawn ball is red is
(a) $$\frac { 3 }{10}$$
(b) $$\frac { 2 }{5}$$
(c) $$\frac { 1 }{5}$$
(d) $$\frac { 3 }{4}$$

Answer: (b) $$\frac { 2 }{5}$$
Hint:
Let E1 : Event that first ball is red
E2 : Event that first ball is black and
E3 : Event that second is the red.
Now, P(E) = P(E1) P |E/E1| + P (E2)P(E/E2)
= $$\frac { 4 }{10}$$ × $$\frac { 6 }{12}$$ + $$\frac { 6 }{10}$$ × $$\frac { 4 }{12}$$
= $$\frac { 1 }{5}$$ + $$\frac { 1 }{5}$$ = $$\frac { 2 }{5}$$

Fill in the blanks

Question 1.
If P(A) = $$\frac { 1 }{5}$$ and P(A – B) = $$\frac { 1 }{6}$$, then P(A∩B) = ………………….

Answer: $$\frac { 1 }{30}$$

Question 2.
The probability of ‘Ace of spade’ is ……………..

Answer: $$\frac { 1 }{52}$$

Question 3.
If P(A) = $$\frac { 6 }{11}$$, P(B) = $$\frac { 5 }{11}$$ and P(A∪B) = then P(B/A) = ……………

Answer: $$\frac { 6 }{11}$$

Question 4.
If A and B are independent events, then P(A∩B) = ………………..

Question 5.
If P($$\bar { A}$$) = 0.4, P(A∪B) = 0.7 and A and B are given to be independent events, then P(B) = ……………..

Answer: $$\frac { 1 }{4}$$

Question 6.
If A and B are two independent events such that P(A) = $$\frac { 1 }{2}$$, P(A∪B) = $$\frac { 3 }{5}$$ and P(A) = p, then p = ………………….

Answer: $$\frac { 1 }{5}$$

Question 7.
If A and B are independent events such that P(A) = $$\frac { 3 }{10}$$, P(B) = $$\frac { 2 }{5}$$ then P(A and B) is ……………….

Answer: $$\frac { 3 }{25}$$

Question 8.
A pair of coins is tossed once. Then the probability of showing at least one head is ………………..

Answer: $$\frac { 3 }{4}$$

Question 9.
A random variable ‘X’ has a probability distribution P(X) of the following form (k is constant) Then k is ……………

Answer: $$\frac { 1 }{6}$$

Question 10.
The mean of the number of heads in the two tosses of a coin is …………………