VBQs Probability Class 12 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 12 Mathematics with solutions. The following Probability Class 12 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 12 examinations.

**Probability VBQs Class 12 Mathematics**

**Question. A four-digit number is formed by the digits 1, 2, 3, 4 with no repetition. The probability that the number is odd, is**

(a) zero

(b) 1/3

(c) 1/4

(d) None of these

**Answer**

D

**Question. A manufacturer of cotter pins knows that 5% of his product is defective. He sells pins in boxes of 100 and guarantees that not more than one pin will be defective in a box. In order to find the probability that a box will fail to meet the guaranteed quality, the probability distribution one has to employ is**

(a) Binomial

(b) Poisson

(c) Normal

(d) Exponential

**Answer**

B

**Question. The probability that atleast one of the events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.2, then P(A̅) + P(B̅) is**

(a) 0.4

(b) 0.8

(c) 1.2

(d) 1.4

**Answer**

C

**Question. A dice is rolled twice and the sum of the numbers appearing on them is observed to be 7. What is the conditional probability that the number 2 has appeared at least once? **

(a) 1/2

(b) 1/3

(c) 2/3

(d) 2/5

**Answer**

B

**Question. If X is a poisson variate such that P(X = 1) = P(X = 2), then P(X = 4) is equal to**

(a) 1/2e^{2}

(b) 1/3e^{2}

(c) 2/3e^{2}

(d) 1/e^{2 }

**Answer**

C

**Question. The probability that a certain kind of component will survive a given shock test is 3/4 . The probability that exactly 2 of the next 4 components tested survive is **

(a) 9/41

(b) 25/128

(c) 1/5

(d) 27/128

**Answer**

D

**Question. If two events are A and B. If odds against A are as 2 : 1 and those in favour of A ∪ B are as 3 : 1, then**

(a) 1/2 ≤ P(B) ≤ 3/4

(b) 5/12 ≤ P(B) ≤ 3/4

(c) 1/4 ≤ P(B) ≤ 3/5

(d) None of these

**Answer**

B

**Question. The chances of defective screws in three boxes A, B, C are 1/5 ,1/6, 1/7 respectively. A box is selected at random and a screw drawn from it at random is found to be defective. Then, the probability that it came from box A, is**

(a) 16/29

(b) 1/15

(c) 27/59

(d) 42/107

**Answer**

D

**Question. If X is a binomial variate with the range {0, 1, 2, 3, 4, 5, 6} and P(X = 2) = 4P(X = 4), then the parameter p of X is**

(a) 1/3

(b) 1/2

(c) 2/3

(d) 3/4

**Answer**

A

**Question. If X follows a binomial distribution with parameters n = 100 and p = 1/3 , then P(X = r) is maximum when r is equal to**

(a) 16

(b) 32

(c) 33

(d) none of these

**Answer**

C

**Question. A random variable X follows binomial distribution with mean a and variance b. Then**

(a) 0 < α < β

(b) 0 < β < α

(c) α < 0 < β

(d) β < 0 < α

**Answer**

B

**Question. If the integers m and n are chosen at random from 1 to 100, then the probability that a number of the form 7n + 7m is divisible by 5, equals to**

(a) 1/4

(b) 1/2

(c) 1/8

(d) 1/3

**Answer**

A

**Question. Mean and standard deviation from the following observations of marks of 5 students of a tutorial group (marks out of 25) 8 12 13 15 22 are**

(a) 14, 4.604

(b) 15, 4.604

(c) 14, 5.604

(d) None of these

**Answer**

A

**Question. If P(A) = 1/12, P(B) = 5/12 and P(B/A) = 1/15 then P(A ∪ B) is equal to **

(a) 89/180

(b) 90/180

(c) 91/180

(d) 92/180

**Answer**

A

**Question. At a telephone enquiry system the number of phone calls regarding relevant enquiry follow Poisson distribution with an average of 5 phone calls during 10 minute time intervals. The probability that there is at the most one phone call during a 10-minute time period is**

(a) 6/5^{e}

(b) 5/6

(c) 6/55

(d) 6/e^{5}

**Answer**

D

**Question. A player X has a biased coin whose probability of showing heads is p and a player Y has a fair coin. They start playing a game with their own coins and play alternately. The player who throws a head first is a winner. If X starts the game, and the probability of winning the game by both the players is equal, then the value of ‘p’ is**

(a) 1/3

(b) 1/5

(c) 1/4

(d) 2/5

**Answer**

A

**Question. It is given that the events A and B are such that P(A) =1/4, P(A | B) = 1/2 and P(B | A) = 2/3, Then P(B) is**

(a) 1/6

(b) 1/3

(c) 2/3

(d) 1/2

**Answer**

B

**Question. Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2, respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is**

(a) 0.2

(b) 0.7

(c) 0.06

(d) 0.14

**Answer**

D

**Question. Three persons P, Q and R independently try to hit a target. If the probabilities of their hitting the target are 3/4, 1/2 and 5/8 respectively, then the probability that the target is hit by P or Q but not by R is :**

(a) 21/64

(b) 9/64

(c) 15/64

(d) 39/64

**Answer**

A

**Question. A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:**

(a) 6/25

(b) 12/5

(c) 6

(d) 4

**Answer**

B

**Question. An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is :**

(a) 496/729

(b) 192/729

(c) 240/729

(d) 256/729

**Answer**

D

**Question. An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :**

(a) 255/256

(b) 127/128

(c) 63/64

(d) 1/2

**Answer**

B

**Question. Let X be a set containing 10 elements and P(X) be its power set. If A and B are picked up at random from P(X), with replacement, then the probability that A and B have equal number elements, is :**

(image 32)

**Answer**

D

**Question. If X has a binomial distribution, B(n, p) with parameters n and p such that P(X = 2) = P (X = 3), then E(X), the mean of variable X, is**

(a) 2 – p

(b) 3 – p

(c) p/2

(d) p/3

**Answer**

B

**Question. A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers ust by guessing is:**

(a) 17/3^{5}

(b) 13/3^{5}

(c) 11/3^{5}

(d) 10/3^{5}

**Answer**

C

**Question. Let A and B be two events such that (image 33) where A̅ stands for the complement of the event A. Then the events A and B are**

(a) independent but not equally likely.

(b) independent and equally likely.

(c) mutually exclusive and independent.

(d) equally likely but not independent.

**Answer**

A

**Question. Given two independent events, if the probability that exactly one of them occurs is 26/49 and the probability that none of them occurs is 15/49 , then the probability of more probable of the two events is :**

(a) 4/7

(b) 6/7

(c) 3/7

(d) 5/7

**Answer**

A

**Question. One ticket is selected at random from 50 tickets numbered 00,01,02,…,49. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is ero, equals:**

(a) 1/7

(b) 5/14

(c) 1/50

(d) 1/14

**Answer**

D

**Question. 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) 2/5

(b) 1/5

(c) 3/4

(d) 3/10

**Answer**

A

**Question. Let A, B and C be three events, which are pair-wise independence and E̅ denotes the complement of an event E. If P (A ∩ B ∩ C) = 0 and P (C) > 0, then P[( A̅ ∩ B̅ ) |C] is equal to.**

(a) P (A) + P ( B̅ )

(b) P ( A̅ ) – P ( B̅ )

(c) P ( A̅ ) – P (B)

(d) P ( A̅ ) + P ( B̅ )

**Answer**

C

**Question. Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is**

(a) 9/2

(b) 9/1

(c) 9/8

(d) 9/7

**Answer**

B

**Question. Let A and B be two events such that (image 45) where A̅ stands for complement of event A. Then events A and B are**

(a) equally likely and mutually exclusive

(b) equally likely but not independent

(c) independent but not equally likely

(d) mutually exclusive and independent

**Answer**

C

**Question. In a box, there are 20 cards, out of which 10 are labelled as A and the remaining 10 are labelled as B. Cards are drawn at random, one after the other and with replacement, till a second A-card is obtained. The probability that the second A-card appears before the third B-card is :**

(a) 9/16

(b) 11/16

(c) 13/16

(d) 15/16

**Answer**

B

**Question. Let A and B be two independent events such that P(A) = 1/3 and P(B) = 1/6. Then, which of the following is TRUE ?**

(a) P(A/B) = 2/3

(b) P(A/B’) = 1/3

(c) P(A’/B’) = 1/3

(d) P(A/(A ∪ B)) = 1/4

**Answer**

B

**Question. The probability that A speaks truth is 4/5, while the probability for B is 3/4. The probability that they contradict each other when asked to speak on a fact is**

(a) 4/5

(b) 1/5

(c) 7/20

(d) 3/20

**Answer**

C

**Question. A problem in mathematics is given to three students A, B, C and their respective probability of solving the problem is 1/2, 1/3 and 1/4. Probability that the problem is solved is**

(a) 3/4

(b) 1/2

(c) 2/3

(d) 1/3

**Answer**

A

**Question. If A and B are any two events such that P(A) = 2/5 and P(A ∩ B) = 3/20, then the conditional probability, P(A | A’ ∪ B’)) , where A’ denotes the complement of A, is equal to :**

(a) 11/20

(b) 5/17

(c) 8/17

(d) 1/4

**Answer**

B

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

(image 51)**Then, P(X > 2) is equal to:**

(a) 7/12

(b) 1/36

(c) 1/6

(d) 23/36

**Answer**

D

**Question. A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If X be the number of white balls drawn, then ( (mean of X)/(standard deviation of X) ) is equal to:**

(a) 4

(b) 4√3

(c) 3√2

(d) 4√3/3

**Answer**

B

**Question. The mean and variance of a random variable X having binomial distribution are 4 and 2 respectively, then P (X = 1) is**

(a) 1/4

(b) 1/32

(c) 1/16

(d) 1/8

**Answer**

B

**Question. Four persons can hit a target correctly with probabilities 1/2, 1/3, 1/4 and 1/8 respectively. If all hit at the target independently, then the probability that the target would be hit, is:**

(a) 25/192

(b) 7/32

(c) 1/192

(d) 25/32

**Answer**

D

**Question. Let A and B be two non-null events such that A ⊂ B. Then, which of the following statements is always correct?**

(a) P(A|B) = P(B) – P(A)

(b) P(A|B) ≥ P(A)

(c) P(A|B) ≤ P(A)

(d) P(A|B) = 1

**Answer**

B

**Question. A dice is tossed 5 times. Getting an odd number is considered a success. Then the variance of distribution of success is**

(a) 8/3

(b) 3/8

(c) 4/5

(d) 5/4

**Answer**

D

**Question. If the mean and the variance of a binomial variate X are 2 and 1 respectively, then the probability that X takes a value greater than or equal to one is :**

(a) 9/16

(b) 3/4

(c) 1/16

(d) 15/16

**Answer**

D

**Question. The probability that a randomly chosen 5-digit number is made from exactly two digits is :**

(a) 135/10^{4}

(b) 121/10^{4}

(c) 150/10^{4}

(d) 134/10^{4}

**Answer**

A

**Question. Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is :**

(a) 2/3

(b) 8/17

(c) 4/17

(d) 2/5

**Answer**

B

**Question. Consider 5 independent Bernoulli’s trials each with probability of success p. If the probability of at least one failure is greater than or equal to 31/32, then p lies in the interval**

(a) (3/4, 11/12]

(b) [0, 1/2]

(c) (11/12, 1]

(d) (1/2, 3/4]

**Answer**

B

**Question. Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is ______.**

Ans : 11

**Question. Let two fair six-faced dice A and B be thrown simultaneously. If E _{1} is the event that die A shows up four, E_{2} is the event that die B shows up two and E_{3} is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true ?**

(a) E

_{1}and E

_{3}are independent.

(b) E

_{1}, E

_{2}and E

_{3}are independent.

(c) E

_{1}and E

_{2}are independent.

(d) E

_{2}and E

_{3}are independent.

**Answer**

B

**Question. In a binomial distribution B(n,p = 1/4), if the probability of at least one success is greater than or equal to 9/10, then n is greater than:**

(a) 1/(log_{10}4+log_{10}3)

(b) 9/(log_{10}4 − log_{10}3)

(c) 4/(log_{10}4 − log_{10}3)

(d) 1/(log_{10}4 − log_{10}3)

**Answer**

D

**Question. A random variable X has Poisson distribution with mean 2. Then P (X > 1.5) equals**

(a) 2/e^{2}

(b) 0

(c) 1 − 3/e^{2}

(d) 3/e^{2}

**Answer**

C

**Question. In a workshop, there are five machines and the probability of any one of them to be out of service on a day is 1/4. If the probability that at most two machines will be out of service on the same day is (3/4) ^{3} k, then k is equal to:**

(a) 17/8

(b) 17/4

(c) 17/2

(d) 4

**Answer**

A

**Question. For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is 4/5, then the probability that he is unable to solve less than two problems is :**

(image 10)

Ans : C

**Question. If two different numbers are taken from the set (0, 1, 2, 3, ……., 10), then the probability that their sum as well as absolute difference are both multiple of 4, is :**

(a) 7/55

(b) 6/55

(c) 12/55

(d) 14/55

**Answer**

B

**Question. Let E and F be two independent events. The probability that both E and F happen is 1/12 and the probability that neither E nor F happens is 1/2, then a value of P(E)/P(F) is :**

(a) 4/3

(b) 3/2

(c) 1/3

(d) 5/12

**Answer**

A

**Question. The probability of a man hitting a target is 2/5. He fires at the target k times (k, a given number). Then the minimum k, so that the probability of hitting the target at least once is more than 7/10, is :**

(a) 3

(b) 5

(c) 2

(d) 4

**Answer**

A

**Question. Three numbers are chosen at random without replacement from {1,2,3,..8}. The probability that their minimum is 3, given that their maximum is 6, is :**

(a) 3/8

(b) 1/5

(c) 1/4

(d) 2/5

**Answer**

B

**Question. If the probability density function of a random variable X is f (x) = x/2 in 0 ≤ x ≤ 2, then P(X > 1.5|X > 1) is equal to**

(a) 7/16

(b) 3/4

(c) 7/12

(d) 21/64

**Answer**

C

**Question. Let X denote the sum of the numbers obtained when two fair dice are rolled. The variance and standard deviation of X are **

**Answer**

B

**Question. A box contains 9 tickets numbered 1 to 9 inclusive. If 3 tickets are drawn from the box one at a time, the probability that they are alternatively either {odd, even, odd} or {even, odd, even} is**

(a) 5/17

(b) 4/17

(c) 5/16

(d) 5/18

**Answer**

D