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Bihar Board 12th Maths Objective Answers Chapter

13 Probability
May 8, 2020 by Shreya

Bihar Board 12th Maths Objective Questions and Answers

Bihar Board 12th Maths Objective Answers Chapter 13

Probability
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﴿ 1

﴾c﴿ 2

﴾d﴿ 1

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

﴾c﴿ 2

﴾d﴿ 1

Answer:
﴾d﴿ 14

Question 4.
If E and F are events such that 0 < P﴾F﴿ < 1, then
﴾a﴿ P (E|F ) + P (Ē |F ) = 1
﴾b﴿ P (E|F ) + P (E|F¯) = 1
﴾c﴿ P (Ē |F ) + P (E|F¯) = 1
﴾d﴿ P (E|F¯) + P (Ē |F¯) = 0
Answer:
﴾a﴿ P (E|F ) + P (Ē |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﴿ 104
21

﴾b﴿ 25

102

﴾c﴿ 23

102

﴾d﴿ 24

104

Answer:
﴾b﴿ 102
25

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﴿ 38
9

﴾b﴿
16

35

﴾c﴿
7

38

﴾d﴿
17

30

Answer:
﴾a﴿ 38
9

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﴿ 15
7

﴾b﴿ 8

15

﴾c﴿ 7

16

﴾d﴿ 5

16

Answer:
﴾b﴿ 15
8
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﴿
3

﴾c﴿ 2

﴾d﴿ 2

Answer:
﴾d﴿ 25

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) =
2

15
and P (A ∩ B̄) = 1

6
,
then find P﴾A﴿ and P ﴾B﴿ respectively.
﴾a﴿ 54 , 45
﴾b﴿ 1

5
,
1

﴾c﴿ 1

6
,
1

﴾d﴿ 1

7
,
1

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’﴿
﴾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﴿ 14
9

﴾b﴿ 7

15

﴾c﴿ 5

14

﴾d﴿ 8

15

Answer:
﴾d﴿ 15
8

Question 15.
If A and Bare events such that P﴾A﴿ = 1

3
, P﴾b﴿ = 1

4
and P﴾A ∩ B﴿ = 1

12
, then find P﴾not A
and not B﴿.
﴾a﴿ 14
﴾b﴿ 1

﴾c﴿ 2

﴾d﴿ 1

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﴿ 1326
103

﴾b﴿
101

1326

﴾c﴿
101

1426

﴾d﴿
103

1426

Answer:
﴾b﴿ 1326
101

Question 17.
Given that, the events A and B are such that P﴾A﴿ = 1

2
, P﴾A ∪ B﴿ = 3

5
and P﴾b﴿ = P. Then
probabilities of B if A and B are mutually exclusive and independent respetively are
﴾a﴿ 1

2
,
1

﴾b﴿ 1

5
,
1

﴾c﴿ 2

3
,
1

﴾d﴿ 1

10
,
1

Answer:
﴾d﴿ 10
1
,
1

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﴿ 4
3

﴾b﴿ 2

﴾c﴿ 1

﴾d﴿ 1

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﴿ 3

﴾c﴿ 7

﴾d﴿ 1

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﴿ 3

﴾c﴿ 4

﴾d﴿ 1

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﴿ 9
5

﴾b﴿ 4

﴾c﴿ 2

﴾d﴿
8

Answer:
﴾c﴿ 29

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﴿ 11
50

﴾b﴿ 17

50

﴾c﴿ 13

50

﴾d﴿ 19

50

Answer:
﴾a﴿ 11
50

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﴿ 1

6
,
11

24
,
33

48

﴾b﴿ 1

6
,
33

48
,
11

24

﴾c﴿ 1

4
,
11

26
,
21

44

﴾d﴿ 11

26
,
1

4
,
21

44

Answer:
﴾b﴿ 16 , 33
48
,
11

24

Question 25.
A coin is tossed until a head appears or the tail appears 4 times in succession. Find the
probability distribution of the number of tosses.

Answer:

Question 26.
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﴿ 221
37

﴾b﴿ 5

13

﴾c﴿ 1

13

﴾d﴿ 2

13

Answer:
﴾d﴿ 13
2

Question 27.
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﴿
3

10

﴾c﴿ 2

﴾d﴿ 1

Answer:
﴾d﴿ 12

Question 28.
The variance and standard deviation of the number of heads in three tosses of a coin
are respectively
√
﴾a﴿ 3 √3
,
4 2

﴾b﴿ 1

4
,
1

﴾c﴿ 3 √3
,
4 4

﴾d﴿ None of these

Answer:
﴾d﴿ None of these

Question 29.
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 , 17
5

﴾b﴿ 13

15
,
15
2

﴾c﴿ 7

10
,
100
21

﴾d﴿ 7

10
,
23

100

Answer:
﴾c﴿ 10
7
,
21

100

Question 30.
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 31.
The variance of random variable X i.e. σx2 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 32.
A coin is biased so that the head is 3 times likely to occur as a tail. If the coin is tossed
twice, then find the probability distribution of the number of tails.

Answer:

Question 33.
A pair of the die is thrown 4 times. If getting a doubled is considered a success, then
find the probability distribution of a number of successes.

Answer:

Question 34.
Find the probability of throwing atmost 2 sixes in 6 throws of a single die.
3
﴾a﴿
35 5
( )
18 6
4
﴾b﴿
35 5
( )
18 6
4
﴾c﴿
18 2
( )
29 3
3
﴾d﴿
18 2
( )
29 3

Answer:
4
﴾b﴿
35 5
( )
18 6

Question 35.
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﴿ 23329
625

﴾b﴿ 621

25329

﴾c﴿ 625

23328

﴾d﴿ 620

23328

Answer:
﴾c﴿ 23328
625

Question 36.
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
10

﴾a﴿ 1 − 7
10
10
10

﴾b﴿ 1 + 7
10
10
10

﴾c﴿ 1 + 9
10
10
10

﴾d﴿ 1 − 9
10
10

Answer:
10

﴾d﴿ 1 − 9
10
10

Question 37.
The probability of a man hitting a target is 1

4
. How many times must he fire so that the
probability of his hitting the target at least once is greater than 2

3
?
﴾a﴿ 4
﴾b﴿ 3
﴾c﴿ 2
﴾d﴿ 1
Answer:
﴾a﴿ 4

Question 38.
Eight coins are thrown simultaneously. Find the probability of getting atleast 6 heads.
﴾a﴿ 128
31

﴾b﴿ 37

256

﴾c﴿ 37

128

﴾d﴿ 31

256

Answer:
﴾b﴿ 256
37

Question 39.
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﴿
3

﴾c﴿
3

﴾d﴿ 1

Answer:
﴾b﴿ 8
3
Question 40.
Eight coins are thrown simultaneously. What is the probability of getting atleast 3
heads?
﴾a﴿ 246
37

﴾b﴿ 21

256

﴾c﴿ 219

256

﴾d﴿ 19

246

Answer:
﴾c﴿ 219
256

Question 41.
If the chance that a ship arrives safely at a port is 9

10
; find the chance that out of 5
expected ships, atleast 4 will arrive safely at the port.
﴾a﴿ 100000
91854

﴾b﴿ 32805

100000

﴾c﴿ 59049

100000

﴾d﴿ 26244

100000

Answer:
﴾a﴿ 100000
91854

Question 42.
If the mean and the variance of a binomial distribution are 4 and, then find P﴾X ≥ 1﴿.
﴾a﴿ 729
720

﴾b﴿
721

729

﴾c﴿
728

729

﴾d﴿
724

729

Answer:
﴾c﴿ 729
728

Question 43.
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﴿ 400

9
,
1600

81

﴾b﴿ 1600

81
,
400

﴾c﴿ 1600

81
,
200

﴾d﴿ 200

9
,
1600

81

Answer:
﴾b﴿ 1600
81
,
400

Question 44.
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 45.
If the sum and the product of the mean and variance of a binomial distribution are 24
and 128 respectively, then find the distribution.
32
﴾a﴿ ( 14
3
+ )
4
30
﴾b﴿ ( 21
+
1

2
)
32
﴾c﴿ ( 12 +
1

2
)
30
﴾d﴿ ( 14
3
+ )
4

Answer:
32
﴾c﴿ ( 12 +
1

2
)

Question 46.
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.
25
﴾a﴿ ( 23 +
1

3
)
25
﴾b﴿ ( 12 +
1

2
)
27
﴾c﴿ ( 12 +
1

2
)
27
﴾d﴿ ( 3 2
+
1

3
)

Answer:
27
﴾d﴿ ( 23 +
1

3
)

Question 47.
The mean and the variance of a binomial distribution are 4 and 2 respectively. Find the
probability of atleast 6 successes.
﴾a﴿ 256
37

﴾b﴿
32

255

﴾c﴿
34

259

﴾d﴿
31

256

Answer:
﴾a﴿ 256
37

Question 48.
If P﴾A ∩ B﴿ = 7

10
and P﴾b﴿ = 17

20
, P﴾A|B﴿ equals
﴾a﴿ 14

17

﴾b﴿ 17

20

﴾c﴿ 7

﴾d﴿ 1

Answer:
﴾a﴿ 14
17

Question 49.
If P﴾A﴿ = 10 , P﴾b﴿ = and P﴾A ∪ B﴿ = , then P﴾B|A﴿ + P﴾A|B﴿ equals
3 2 3

5 5

﴾a﴿ 1

﴾b﴿ 1

﴾c﴿
5

12

﴾d﴿
7

12

Answer:
﴾d﴿ 12
7
Question 50.
If P﴾A﴿ = 25 , P﴾B﴿ = and P﴾A ∩ B﴿ = , then P﴾A’|B’﴿ . ﴾P﴾B’|A’﴿ is equal to
3 1

10 5

﴾a﴿
5

﴾b﴿
5

﴾c﴿
25

42

﴾d﴿ 1
Answer:
﴾b﴿ 7
5

Question 51.
If A and B are two events sue that P﴾A﴿ = 1

2
, P﴾b﴿ = 1

3
, P﴾A|B﴿ = 1

4
then ﴾A’ ∩ B’﴿ equals
﴾a﴿ 1

12

﴾b﴿ 3

﴾c﴿ 1

﴾d﴿ 3

16

Answer:
﴾c﴿ 14

Question 52.
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 53.
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 54.
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
equals
﴾a﴿ 23
﴾b﴿ 1

﴾c﴿
3

10

﴾d﴿ 1

Answer:
﴾d﴿ 15

Question 55.
You are given that A and B are two events such that P﴾b﴿ = , P﴾A|B﴿ = = , then P﴾A﴿
3 4

5 5

equals
﴾a﴿ 10
3

﴾b﴿ 1

﴾c﴿ 1

﴾d﴿
3

Answer:
﴾c﴿ 12

Question 56.
If P﴾b﴿ = 35 , P﴾A|B﴿ = 1

2
and P﴾A ∪ B﴿ = 4

5
, then P﴾A ∪ B’﴿ + P﴾A’ ∪ B﴿ = 1
﴾a﴿ 1

﴾b﴿ 4

﴾c﴿ 1

﴾d﴿ 1
Answer:
﴾d﴿ 1

Question 57.
If A and Bare two independent events with P﴾A﴿ = and P﴾b﴿ = , then P﴾A’ ∩ B’﴿
3 4

5 9

equals
﴾a﴿ 15
4

﴾b﴿
8

45

﴾c﴿ 1

﴾d﴿ 2

Answer:
﴾d﴿ 29
Question 58.
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 59.
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﴿ 3

35

﴾c﴿ 1

70

﴾d﴿ 1

Answer:
﴾c﴿ 70
1

Question 60.
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﴿ 196
45

﴾b﴿
135

392

﴾c﴿
15

56

﴾d﴿
15

29

Answer:
﴾c﴿ 56
15

Question 61.
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﴿ 1

﴾b﴿ 1

﴾c﴿ 1

﴾d﴿ 3

Answer:
﴾c﴿ 18

Question 62.
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﴿ 28
3

﴾b﴿ 2

21

﴾c﴿ 1

28

﴾d﴿
167

168

Answer:
﴾a﴿ 28
3

Question 63.
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﴿ 33
56

﴾b﴿ 9

64

﴾c﴿ 1

14

﴾d﴿ 3

28

Answer:
﴾d﴿ 28
3

Question 64.
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﴿ 18
1

﴾b﴿ 5

18

﴾c﴿ 1

﴾d﴿ 2

Answer:
﴾c﴿ 1

Question 65.
Two cards are drawn from a well shuffled deck of 52 playing cards with replacement.
The probability, that both cards are queens, is
﴾a﴿ 13
1
×
1

13

﴾b﴿ 1

13
+
1

13

﴾c﴿ 1

13
×
1

17

﴾d﴿ 1

13
×
4

51

Answer:
﴾a﴿ 13
1
×
1

13

Question 66.
The probability of guessing correctly at least 8 out of 10 answers on a true‐false type
examiniation is
﴾a﴿ 64
7

﴾b﴿ 128
7

﴾c﴿ 45

1024

﴾d﴿ 7

41

Answer:
﴾b﴿ 128
7

Question 67.
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 68.
For the following probability distribution:

E﴾X﴿ is equal to
﴾a﴿ 0
﴾b﴿ ‐1
﴾c﴿ ‐2
﴾d﴿ ‐1.8
Answer:
﴾d﴿ ‐1.8

Question 69.
For the following probability distribution

E﴾X2﴿ is equal to
﴾a﴿ 3
﴾b﴿ 5
﴾c﴿ 7
﴾d﴿ 10
Answer:
﴾d﴿ 10

Question 70.
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﴿ 1

﴾c﴿ 1

﴾d﴿ 1

Answer:
﴾a﴿ 12

Question 71.
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?
5
﴾a﴿ ( 10
9
)
4
﴾b﴿ 1

2
(
9

10
)
5
﴾c﴿ 1

2
(
9

10
)
5 4
﴾d﴿ ( 10 )9
+
1

2
(
9

10
)

Answer:
5 4
﴾d﴿ ( 10
9
) +
1

2
(
9

10
)
 Class 12
 Bihar Board Class 12th Hindi रचना िनबं
ध ले
खन
 Bihar Board 12th English 100 Marks Objective Answers Chapter 2 Bharat is My Home

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