12th Math chapter 13: full solution, notes, MCQ, & free PDF | Bihar board

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12th Math chapter 13 Bihar board with NCERT-Details & Highlights

name of content	12th Math chapter 13 full solution
content	Math objective type MCQ question with answer
name of subjects	class 12 Math
important for exam	Bihar board & other state board
pattern & Level	NCERT & State level question

12th Math chapter 13 all objective solutions – download PDF

Question 1. P has 2 children. He has a son, Jatin. What is the probability that Jatin’s sibling is a brother?

• • (a) 13
    (b) 14
    (c) 23
    (d) 12

Answer: (a) 13

Question 2. If A and B are 2 events such that P(A) > 0 and P (b) ≠ 1, then P(A¯/B¯)=

• • (a) 1 – P(A|B)
    (b) 1−P(A/B¯)
    (c) 1−P(A∪B)P(B)
    (d) 1(A¯)P(B)

Answer: (b) 1−P(A/B¯)

Question 3. If two events A and B area such that P(A¯) =0.3, P(B) = 0.4 and P(B|A∪B¯)=

• • (a) 12
    (b) 13
    (c) 25
    (d) 14

Answer: (d) 14

Question 4. If E and F are events such that 0 < P(F) < 1, then

• • (a) P(E|F)+P(E¯|F)=1
    (b) P(E|F)+P(E|F¯)=1
    (c) P(E¯|F)+P(E|F¯)=1
    (d) P(E|F¯)+P(E¯|F¯)=0

Answer: (a) P(E|F)+P(E¯|F)=1

Question 5. P(E ∩ F) is equal to

• • (a) P(E) . P(F|E)
    (b) P(F) . P(E|F)
    (c) Both (a) and (b)
    (d) None of these

Answer: (c) Both (a) and (b)

Question 6. If three events of a sample space are E, F and G, then P(E ∩ F ∩ G) is equal to

• • (a) P(E) P(F|E) P(G|(E ∩ F))
    (b) P(E) P(F|E) P(G|EF)
    (c) Both (a) and (b)
    (d) None of these

Answer: (c) Both (a) and (b)

Question 7. Two cards are drawn at random one by one without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.

• • (a) 21104
    (b) 25102
    (c) 23102
    (d) 24104

Answer: (b) 25102

Question 8. A bag contains 20 tickets, numbered 1 to 20. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will show even numbers.

• • (a) 938
    (b) 1635
    (c) 738
    (d) 1730

Answer: (a) 938

Question 9. Two balls are drawn one after another (without replacement) from a bag containing 2 white, 3 red and 5 blue balls. What is the probability that atleast one ball is red?

• • (a) 715
    (b) 815
    (c) 716
    (d) 516

Answer: (b) 815

Question 10. Let A and B be independent events with P(A) = 1/4 and P(A ∪ B) = 2P(B) – P(A). Find P(B)

• • (a) 14
    (b) 35
    (c) 23
    (d) 25

Answer: (d) 25

12th math chapter 13 oblective solution

Question 11. Two events A and B will 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: (c) P(A) = P(B)

Question 12. If A and B are two independent events such that P(A¯∩B)=215 and P(A∩B¯)=16, then find P(A) and P (B) respectively.

• • (a) 54,45
    (b) 15,17
    (c) 16,17
    (d) 17,17

Answer: (a) 54,45

Question 13. If A and B are two independent events, then the probability of occurrence of at least of A and B is given by

• • (a) 1 – P(A) P(b)
    (b) 1 – P(A) P(B’)
    (c) 1 – P(A’) P(B’)
    (d) 1 – P(A’) P(b)

Answer: (c) 1 – P(A’) P(B’)

Question 14. If A and B are two indendent events such that P(A¯) = 0.75, P(A ∪ B) = 0.65 and P(b) = P, then find the value of P.

• • (a) 914
    (b) 715
    (c) 514
    (d) 815

Answer: (d) 815

Question 15. If A and Bare events such that P(A) = 13, P(b) = 14 and P(A ∩ B) = 112, then find P(not A and not B).

• • (a) 14
    (b) 12
    (c) 23
    (d) 13

Answer: (b) 12

Question 16. Two cards are drawn successively from a well shuffled pack of 52 cards. Find the probability that one is a red card the other is a queen.

• • (a) 1031326
    (b) 1011326
    (c) 1011426
    (d) 1031426

Answer: (b) 1011326

Question 17. Given that, the events A and B are such that P(A) = 12, P(A ∪ B) = 35 and P(b) = P. Then probabilities of B if A and B are mutually exclusive and independent respetively are

• • (a) 12,13
    (b) 15,13
    (c) 23,13
    (d) 110,15

Answer: (d) 110,15

Question 18. Two cards from an ordinary deck of 52 cards are missing. What is the probability that a random card drawn from this deck is a spade?

• • (a) 34
    (b) 23
    (c) 12
    (d) 14

Answer: (d) 14

Question 19. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

• • (a) 58
    (b) 38
    (c) 78
    (d) 18

Answer: (b) 38

Question 20. A bag contains 4 balls. Two balls are drawn at random and are found to be white. What is the probability that all balls are white?

• • (a) 25
    (b) 35
    (c) 45
    (d) 15

Answer: (b) 35

Question 21. A bag contains 3 green and 7 white balls. Two balls are drawn one by one at random without replacement. If the second ball drawn is green, what is the probability that the first ball was drawn in also green?

• • (a) 59
    (b) 49
    (c) 29
    (d) 89

Answer: (c) 29

Bihar Board most mcq solution 12th math chapter 13

Question 22. A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both clubs. Find the probability of the lost card being a club.

• • (a) 1150
    (b) 1750
    (c) 1350
    (d) 1950

Answer: (a) 1150

Question 23. A random variable X has the following distribution.

For the event E = {X is prime number} and F = {X < 4}, P(E ∪ F) =

• • (a) 0.87
    (b) 0.77
    (c) 0.35
    (d) 0.50

Answer: (b) 0.77

Question 24. A random variable X has the following probability distribution:

Find P(X < 3), P(X ≥ 4), P(0 < X < 5) respectively.

• • (a) 16,1124,3348
    (b) 16,3348,1124
    (c) 14,1126,2144
    (d) 1126,14,2144

Answer: (b) 16,3348,1124

Question 25. Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. Then, the value of E(X) is

• • (a) 37221
    (b) 513
    (c) 113
    (d) 213

Answer: (d) 213

Question 26. The random variable X can take only the values 0, 1, 2. Given that, P(X = 0) = P (X = 1) = p and that E(X²) = E(X), find the value of p.

• • (a) 15
    (b) 310
    (c) 25
    (d) 12

Answer: (d) 12

Question 27. The variance and standard deviation of the number of heads in three tosses of a coin are respectively

• • (a) 34,3√2
    (b) 14,12
    (c) 34,3√4
    (d) None of these

Answer: (d) None of these

Question 28. In a meeting, 70% of the members favour and 30% oppose a certain proposal, A member is selected at random and we take X = 0, if opposed and X = 1, if he is in favour. Then, E(X) and Var(X) are respectively

• • (a) 37,517
    (b) 1315,215
    (c) 710,21100
    (d) 710,23100

Answer: (c) 710,21100

Question 29. For the following probability distribution, the standard deviation of the random variable X is

• • (a) 0.5
    (b) 0.6
    (c) 0.61
    (d) 0.7

Answer: (d) 0.7

Question 30. The variance of random variable X i.e. σ2x or var (X) is equal to

• • (a) E(X²) + [E(X²)₂]²
    (b) E(X) – [E(X²)] (c) E(X²) – [E(X)]²
    (d) None of these

Answer: (c) E(X²) – [E(X)]²

Question 31. Find the probability of throwing atmost 2 sixes in 6 throws of a single die.

• • (a) 3518(56)3
    (b) 3518(56)4
    (c) 1829(23)4
    (d) 1829(23)3Answer:

Answer: (b) 3518(56)4

Question 32. A die is thrown again and again until three sixes are obtained. Find the probability of obtaining third six in the sixth throw of the die.

• • (a) 62523329
    (b) 62125329
    (c) 62523328
    (d) 62023328

Answer: (c) 62523328

BSEB 12th math chapter 13 most mcq notes 

Question 33. Ten eggs are drawn successively with replacement from a lot containing 10% defective eggs. Then, the probability that there is atleast one defective egg is

• • (a) 1−7101010
    (b) 1+7101010
    (c) 1+9101010
    (d) 1−9101010

Answer: (d) 1−9101010

Question 34. The probability of a man hitting a target is 14. How many times must he fire so that the probability of his hitting the target at least once is greater than 23?

• • (a) 4
    (b) 3
    (c) 2
    (d) 1

Answer: (a) 4

Question 35. Eight coins are thrown simultaneously. Find the probability of getting atleast 6 heads.

• • (a) 31128
    (b) 37256
    (c) 37128
    (d) 31256

Answer: (b) 37256

Question 36. A bag contains 6 red, 4 blue and 2 yellow balls. Three balls are drawn one by one with replacement. Find the probability of getting exactly one red ball.

• • (a) 14
    (b) 38
    (c) 34
    (d) 12

Answer: (b) 38

Question 37. Eight coins are thrown simultaneously. What is the probability of getting atleast 3 heads?

• • (a) 37246
    (b) 21256
    (c) 219256
    (d) 19246

Answer: (c) 219256

Question 38. If the chance that a ship arrives safely at a port is 910; find the chance that out of 5 expected ships, atleast 4 will arrive safely at the port.

• • (a) 91854100000
    (b) 32805100000
    (c) 59049100000
    (d) 26244100000

Answer: (a) 91854100000

Question 39. If the mean and the variance of a binomial distribution are 4 and, then find P(X ≥ 1).

• • (a) 720729
    (b) 721729
    (c) 728729
    (d) 724729

Answer: (c) 728729

Question 40. A pair of dice is thrown 200 times. If getting a sum of 9 is considered a success, then find the mean and the variance respectively of the number of successes.

• • (a) 4009,160081
    (b) 160081,4009
    (c) 160081,2009
    (d) 2009,160081

Answer: (b) 160081,4009

Question 41. In a binomial distribution, the sum of its mean and variance is 1.8. Find the probability of two successes, if the event was conducted times.

• • (a) 0.2623
    (b) 0.2048
    (c) 0.302
    (d) 0.305

Answer: (b) 0.2048

Question 42. If the sum and the product of the mean and variance of a binomial distribution are 24 and 128 respectively, then find the distribution.

• • (a) (14+34)32
    (b) (12+12)30
    (c) (12+12)32
    (d) (14+34)30

Answer: (c) (12+12)32

Question 43. If the sum of the mean and variance of a binomial distribution is 15 and the sum of their squares is 17, then find the distribution.

• • (a) (23+13)25
    (b) (12+12)25
    (c) (12+12)27
    (d) (23+13)27

Answer: (d) (23+13)27

Important mcq solution 12th math chapter 13 

Question 44. The mean and the variance of a binomial distribution are 4 and 2 respectively. Find the probability of atleast 6 successes.

• • (a) 37256
    (b) 32255
    (c) 34259
    (d) 31256

Answer: (a) 37256

Question 45. If P(A ∩ B) = 710 and P(b) = 1720, P(A|B) equals

• • (a) 1417
    (b) 1720
    (c) 78
    (d) 18

Answer: (a) 1417

Question 46. If P(A) = 310, P(b) = 25 and P(A ∪ B) = 35, then P(B|A) + P(A|B) equals

• • (a) 14
    (b) 13
    (c) 512
    (d) 712

Answer: (d) 712

Question 47. If P(A) = 25, P(B) = 310 and P(A ∩ B) = 15, then P(A’|B’) . (P(B’|A’) is equal to

• • (a) 56
    (b) 57
    (c) 2542
    (d) 1

Answer: (b) 57

Question 48. If A and B are two events sue that P(A) = 12, P(b) = 13, P(A|B) = 14 then (A’ ∩ B’) equals

• • (a) 112
    (b) 34
    (c) 14
    (d) 316

Answer: (c) 14

Question 49. If P(A) = 0.4, P(b) = 0.8 and P(B|A) = 0.6, then P(A ∪ B) equal to

• • (a) 0.24
    (b) 0.3
    (c) 0.48
    (d) 0.96

Answer: (c) 0.48

Question 50. If A and B are two events and A ≠ Φ, B ≠ Φ, then

• • (a) P(A|B) = P(A) . P(b)
    (b) P(A|B) = P(A∩B)P(B)
    (c) P(A|B) . P(B|A) = 1
    (d) P(A|B) = P(A)|P(b)

Answer: (b) P(A|B) = P(A∩B)P(B)

Question 51. A and B are events such that P(A) = 0.4, P(b) = 0.3 and P(A ∪ B) = 0.5. Then P(B’ ∩ A) equals

• • (a) 23
    (b) 12
    (c) 310
    (d) 15

Answer: (d) 15

Question 52. You are given that A and B are two events such that P(b) = 35, P(A|B) = =45, then P(A) equals

• • (a) 310
    (b) 15
    (c) 12
    (d) 35

Answer: (c) 12

Question 53. If P(b) = 35, P(A|B) = 12 and P(A ∪ B) = 45, then P(A ∪ B’) + P(A’ ∪ B) = 1

• • (a) 15
    (b) 45
    (c) 12
    (d) 1

Answer: (d) 1

Question 54. If A and Bare two independent events with P(A) = 35 and P(b) = 49, then P(A’ ∩ B’) equals

• • (a) 415
    (b) 845
    (c) 13
    (d) 29

Answer: (d) 29

most mcq notes and solution 12th math chapter 13

Question 55. If the events A and B are independet, then P(A ∩ B) is equal to

• • (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 56. Two events E and F are independent. If P(E) = 0.3, P(E ∪ F) = 0.5, then P(E|F) – P(F|E) equals

• • (a) 27
    (b) 335
    (c) 170
    (d) 17

Answer: (c) 170

Question 57. A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replecement the probability of getting exactly one red ball is

• • (a) 45196
    (b) 135392
    (c) 1556
    (d) 1529

Answer: (c) 1556

Question 58. A die is thrown and card is selected a random from a deck of 52 playing cards. The probability of gettingan even number on the die and a spade card is

• • (a) 12
    (b) 14
    (c) 18
    (d) 34

Answer: (c) 18

Question 59. 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) 328
    (b) 221
    (c) 128
    (d) 167168

Answer: (a) 328

Question 60. 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 deal is

• • (a) 3356
    (b) 964
    (c) 114
    (d) 328

Answer: (d) 328

Question 61. Two dice are thrown. If it is known that the sum of numbers on the dice was less than 6, the probability of getting a sum 3, is

• • (a) 118
    (b) 518
    (c) 15
    (d) 25

Answer: (c) 15

Question 62. Two cards are drawn from a well shuffled deck of 52 playing cards with replacement. The probability, that both cards are queens, is

• • (a) 113×113
    (b) 113+113
    (c) 113×117
    (d) 113×451

Answer: (a) 113×113

Question 63. The probability of guessing correctly at least 8 out of 10 answers on a true-false type examiniation is

• • (a) 764
    (b) 7128
    (c) 451024
    (d) 741

Answer: (b) 7128

Question 64. 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 65. For the following probability distribution:

E(X) is equal to

• • (a) 0
    (b) -1
    (c) -2
    (d) -1.8

Answer: (d) -1.8

Question 66. For the following probability distribution

E(X²) is equal to

• • (a) 3
    (b) 5
    (c) 7
    (d) 10

Answer: (d) 10

Question 67. Suppose a random variable X follows the binomial distribution with parameters n and p, where 0 < p < 1. If p(x = r) / P(x = n – r) is dindependent of n and r, then p equals

• • (a) 12
    (b) 13
    (c) 15
    (d) 17

Answer: (a) 12

Question 68. A box has 100 pens of which 10 are defective. What is the probability that out of a sample of 5 pens drawn one by one with replacement at most one is defective?

• • (a) (910)5
    (b) 12(910)4
    (c) 12(910)5
    (d) (910)5+12(910)4

Answer: (d) (910)5+12(910)4

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