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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.

## Answer

Answer: (c) not defined

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

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

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

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

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

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

## Answer

Answer: (a) A ⊂ B

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

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

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

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

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.

## Answer

Answer: (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

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

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).

## Answer

Answer: (b) 4

Question 16.

Let ‘X’ be a discrete random variable assuming values x_{1}, x_{2}, …………… , x_{n} with probabilities p_{1}, p_{2}, …………. , p_{n} respectively. Then variance of ‘X’ is given by

(a) E(X²)

(b) E(X²) + E(X)

(c) E(X²) – [E(X)]²

(d) \(\sqrt { E(X)^2-[E(X)]^2}\)

## Answer

Answer: (c) E(X²) – [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

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.

## Answer

Answer: (b) 0.25

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

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 (A^{c}∩B^{c} /C) is

(a) P (A) – P (B^{c})

(b) P (A^{c}) + P (B^{c})

(c) P (A^{c}) – P (B^{c})

(d) P (A^{c}) – P (B).

## Answer

Answer: (d) P (A^{c}) – 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

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

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

Answer: (c) \(\frac { 6 }{55}\)

Hint:

Let A = {0, 1, 2, 3, …….. , 10}

∴ n(S) = ^{11}C_{2} = 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

Answer: (b) \(\frac { 2 }{5}\)

Hint:

Let E_{1} : Event that first ball is red

E_{2} : Event that first ball is black and

E_{3} : Event that second is the red.

Now, P(E) = P(E_{1}) P |E/E_{1}| + P (E_{2})P(E/E_{2})

= \(\frac { 4 }{10}\) × \(\frac { 6 }{12}\) + \(\frac { 6 }{10}\) × \(\frac { 4 }{12}\)

= \(\frac { 1 }{5}\) + \(\frac { 1 }{5}\) = \(\frac { 2 }{5}\)

Fill in the blanks

Question 1.

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

## Answer

Answer: \(\frac { 1 }{30}\)

Question 2.

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

## Answer

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

Answer: \(\frac { 6 }{11}\)

Question 4.

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

## Answer

Answer: P(A) P(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

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

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

Answer: \(\frac { 3 }{25}\)

Question 8.

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

## Answer

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

Answer: \(\frac { 1 }{6}\)

Question 10.

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

## Answer

Answer: 1.

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