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Probability Class 12 MCQs Questions with Answers
Students are advised to solve the Probability Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Probability Class 12 with answers will boost your confidence thereby helping you score well in the exam.
Explore numerous MCQ Questions of Probability Class 12 with answers provided with detailed solutions by looking below.
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 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) \(\sqrt { E(X)^2-[E(X)]^2}\)
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 (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
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
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) = 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
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
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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