CBSE Class 12 Mathematics Probability MCQs Set 06

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

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. You are given that \(A\) and \(B\) are two events such that \(P(B) = \frac{3}{5}, P(A \mid B) = \frac{1}{2}\) 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. Three persons, A, B and C, fire at a target in turn, starting with A. Their probability of hitting the target are 0.4, 0.3 and 0.2 respectively. The probability of two hits is
(a) 0.024
(b) 0.188
(c) 0.336
(d) 0.452
Answer: (b) 0.188

Question. Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has at least one girl is
(a) \(\frac{1}{2}\)
(b) \(\frac{1}{3}\)
(c) \(\frac{2}{3}\)
(d) \(\frac{4}{7}\)
Answer: (d) \(\frac{4}{7}\)

Question. A die is thrown and a card is selected at random from a deck of 52 playing cards. The probability of getting an even number on the die and a spade card is
(a) \(\frac{1}{2}\)
(b) \(\frac{1}{4}\)
(c) \(\frac{1}{8}\)
(d) \(\frac{3}{4}\)
Answer: (c) \(\frac{1}{8}\)

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. 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. If two events are independent then
(a) they must be mutually exclusive
(b) they sum of their probabilities must be equal to 1
(c) both (a) and (b) are correct.
(d) None of the options
Answer: (d) None of the options

Question. Two dice are thrown. If it is known that the sum of number on the dice was less than 6, the probability of getting a sum 3, is
(a) \(\frac{1}{18}\)
(b) \(\frac{5}{18}\)
(c) \(\frac{1}{5}\)
(d) \(\frac{2}{5}\)
Answer: (c) \(\frac{1}{5}\)

Question. If the events \(A\) and \(B\) are independent, then \(P(A \cap B)\) equals
(a) \(P(A) + P(B)\)
(b) \(P(A) - P(B)\)
(c) \(P(A) . P(B)\)
(d) \(P(A) / P(B)\)
Answer: (c) \(P(A) . P(B)\)

Question. Two cards are drawn from a well shuffled deck of 52 playing cards with replacement. The probability, that both cards are queens, is
(a) \(\frac{1}{13} \times \frac{1}{13}\)
(b) \(\frac{1}{13} \times \frac{1}{12}\)
(c) \(\frac{1}{13} \times \frac{1}{17}\)
(d) \(\frac{1}{13} \times \frac{4}{51}\)
Answer: (a) \(\frac{1}{13} \times \frac{1}{13}\)

Question. A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement, then the probability of getting exactly one red ball is
(a) \(\frac{15}{196}\)
(b) \(\frac{131}{392}\)
(c) \(\frac{15}{56}\)
(d) \(\frac{15}{29}\)
Answer: (c) \(\frac{15}{56}\)

Question. Three persons A, B and C, fire at a target in turn, standing with A. Their probability of hitting the target are 0.4, 0.3 and 0.2 respectively. The probability of two hits is
(a) 0.025
(b) 0.1888
(c) 0.339
(d) 0.475
Answer: (b) 0.1888

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. For the following probability distribution:

\(E(X^2)\) is equal to
(a) 3
(b) 5
(c) 7
(d) 10
Answer: (d) 10

Question. Two dice are thrown together. Let A be the event 'getting 6 on the first die' are B be the event 'getting 2 on the second die', then \(P(A \cap B)\) is
(a) \(\frac{1}{36}\)
(b) \(\frac{7}{4}\)
(c) \(\frac{9}{20}\)
(d) None of the options
Answer: (a) \(\frac{1}{36}\)

Question. In a college, 30% students fail in Physics, 25% fail in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in Physics if she has failed in Mathematics is
(a) \(\frac{1}{10}\)
(b) \(\frac{2}{5}\)
(c) \(\frac{9}{20}\)
(d) \(\frac{1}{3}\)
Answer: (b) \(\frac{2}{5}\)

Question. A and B are two students. Their chances of solving a problem correctly are \(\frac{1}{3}\) and \(\frac{1}{4}\), respectively. If the probability of their making a common error is, \(\frac{1}{20}\) and they obtain the same answer, then the probability of their answer to be correct is 
(a) \(\frac{1}{12}\)
(b) \(\frac{1}{40}\)
(c) \(\frac{13}{120}\)
(d) \(\frac{10}{13}\)
Answer: (d) \(\frac{10}{13}\)

Question. Let X be a discrete random variable assuming values \(x_1, x_2, \dots, x_n\) with probabilities \(p_1, p_2, \dots, p_n\) respectively. Then variance of X is given by 
(a) \(E(X^2)\)
(b) \(E(X^2) + E(X)\)
(c) \(E(X^2) - [E(X)]^2\)
(d) \(\sqrt{E(X^2) - [E(X)]^2}\)
Answer: (c) \(E(X^2) - [E(X)]^2\)

Assertion-Reason Questions

The following questions consist of two statements—Assertion(A) and Reason(R). Answer these questions selecting the appropriate option given below:
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : The probability of obtaining an even prime number on each die, when a pair of dice is rolled is \(\frac{1}{36}\).
Reason (R) : If \(P(A / B) > P(A)\), then \(P(B / A) > P(B)\).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.

Question. Assertion (A) : If \(P(A) = \frac{1}{2}, P(A \cap B) = \frac{1}{3}\), Then the value of \(P(B / A) = \frac{2}{3}\).
Reason (R) : \(P(B / A) = \frac{P(A \cap B)}{P(A)}\).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.

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

then \(a = \frac{1}{81}\).
Reason (R) : The sum of probabilities of a probability distribution is always 1
i.e., \(\Sigma P(X) = 1\).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.

Question. Assertion (A) : If \(P(A) = 0.2, P(B) = 0.3\), and A and B are independent events then \(P(A \cap B) = 0.06\).
Reason (R) : When A and B are independent events than \(P(A \cap B) = P(A) . P(B)\).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.

Question. Assertion (A) : If \(P(A) = \frac{3}{8}\) and \(P(B) = \frac{5}{8}, P(A \cup B) = \frac{3}{4}\) then \(P(A' / B') = \frac{2}{3}\).
Reason (R) : \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.