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1 | 6918-6921 | P (F)
Remarks
(i)
Two events E and F are said to be dependent if they are not independent, i e if
P(E ∩ F ) ≠ P(E) P (F)
(ii)
Sometimes there is a confusion between independent events and mutually
exclusive events |
1 | 6919-6922 | e if
P(E ∩ F ) ≠ P(E) P (F)
(ii)
Sometimes there is a confusion between independent events and mutually
exclusive events Term ‘independent’ is defined in terms of ‘probability of events’
whereas mutually exclusive is defined in term of events (subset of sample space) |
1 | 6920-6923 | if
P(E ∩ F ) ≠ P(E) P (F)
(ii)
Sometimes there is a confusion between independent events and mutually
exclusive events Term ‘independent’ is defined in terms of ‘probability of events’
whereas mutually exclusive is defined in term of events (subset of sample space) Moreover, mutually exclusive events never have an outcome common, but
independent events, may have common outcome |
1 | 6921-6924 | P (F)
(ii)
Sometimes there is a confusion between independent events and mutually
exclusive events Term ‘independent’ is defined in terms of ‘probability of events’
whereas mutually exclusive is defined in term of events (subset of sample space) Moreover, mutually exclusive events never have an outcome common, but
independent events, may have common outcome Clearly, ‘independent’ and
‘mutually exclusive’ do not have the same meaning |
1 | 6922-6925 | Term ‘independent’ is defined in terms of ‘probability of events’
whereas mutually exclusive is defined in term of events (subset of sample space) Moreover, mutually exclusive events never have an outcome common, but
independent events, may have common outcome Clearly, ‘independent’ and
‘mutually exclusive’ do not have the same meaning In other words, two independent events having nonzero probabilities of occurrence
can not be mutually exclusive, and conversely, i |
1 | 6923-6926 | Moreover, mutually exclusive events never have an outcome common, but
independent events, may have common outcome Clearly, ‘independent’ and
‘mutually exclusive’ do not have the same meaning In other words, two independent events having nonzero probabilities of occurrence
can not be mutually exclusive, and conversely, i e |
1 | 6924-6927 | Clearly, ‘independent’ and
‘mutually exclusive’ do not have the same meaning In other words, two independent events having nonzero probabilities of occurrence
can not be mutually exclusive, and conversely, i e two mutually exclusive events
having nonzero probabilities of occurrence can not be independent |
1 | 6925-6928 | In other words, two independent events having nonzero probabilities of occurrence
can not be mutually exclusive, and conversely, i e two mutually exclusive events
having nonzero probabilities of occurrence can not be independent (iii)
Two experiments are said to be independent if for every pair of events E and F,
where E is associated with the first experiment and F with the second experiment,
the probability of the simultaneous occurrence of the events E and F when the
two experiments are performed is the product of P(E) and P(F) calculated
separately on the basis of two experiments, i |
1 | 6926-6929 | e two mutually exclusive events
having nonzero probabilities of occurrence can not be independent (iii)
Two experiments are said to be independent if for every pair of events E and F,
where E is associated with the first experiment and F with the second experiment,
the probability of the simultaneous occurrence of the events E and F when the
two experiments are performed is the product of P(E) and P(F) calculated
separately on the basis of two experiments, i e |
1 | 6927-6930 | two mutually exclusive events
having nonzero probabilities of occurrence can not be independent (iii)
Two experiments are said to be independent if for every pair of events E and F,
where E is associated with the first experiment and F with the second experiment,
the probability of the simultaneous occurrence of the events E and F when the
two experiments are performed is the product of P(E) and P(F) calculated
separately on the basis of two experiments, i e , P (E ∩ F) = P (E) |
1 | 6928-6931 | (iii)
Two experiments are said to be independent if for every pair of events E and F,
where E is associated with the first experiment and F with the second experiment,
the probability of the simultaneous occurrence of the events E and F when the
two experiments are performed is the product of P(E) and P(F) calculated
separately on the basis of two experiments, i e , P (E ∩ F) = P (E) P(F)
(iv)
Three events A, B and C are said to be mutually independent, if
P(A ∩ B) = P(A) P(B)
P(A ∩ C) = P(A) P(C)
P(B ∩ C) = P(B) P(C)
and
P(A ∩ B ∩ C) = P(A) P(B) P(C)
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544
MATHEMATICS
If at least one of the above is not true for three given events, we say that the
events are not independent |
1 | 6929-6932 | e , P (E ∩ F) = P (E) P(F)
(iv)
Three events A, B and C are said to be mutually independent, if
P(A ∩ B) = P(A) P(B)
P(A ∩ C) = P(A) P(C)
P(B ∩ C) = P(B) P(C)
and
P(A ∩ B ∩ C) = P(A) P(B) P(C)
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544
MATHEMATICS
If at least one of the above is not true for three given events, we say that the
events are not independent Example 10 A die is thrown |
1 | 6930-6933 | , P (E ∩ F) = P (E) P(F)
(iv)
Three events A, B and C are said to be mutually independent, if
P(A ∩ B) = P(A) P(B)
P(A ∩ C) = P(A) P(C)
P(B ∩ C) = P(B) P(C)
and
P(A ∩ B ∩ C) = P(A) P(B) P(C)
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544
MATHEMATICS
If at least one of the above is not true for three given events, we say that the
events are not independent Example 10 A die is thrown If E is the event ‘the number appearing is a multiple of
3’ and F be the event ‘the number appearing is even’ then find whether E and F are
independent |
1 | 6931-6934 | P(F)
(iv)
Three events A, B and C are said to be mutually independent, if
P(A ∩ B) = P(A) P(B)
P(A ∩ C) = P(A) P(C)
P(B ∩ C) = P(B) P(C)
and
P(A ∩ B ∩ C) = P(A) P(B) P(C)
© NCERT
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544
MATHEMATICS
If at least one of the above is not true for three given events, we say that the
events are not independent Example 10 A die is thrown If E is the event ‘the number appearing is a multiple of
3’ and F be the event ‘the number appearing is even’ then find whether E and F are
independent Solution We know that the sample space is S = {1, 2, 3, 4, 5, 6}
Now
E = { 3, 6}, F = { 2, 4, 6} and E ∩ F = {6}
Then
P(E) = 2
1
3
1
1
, P(F)
and P(E
F)
6
3
6
2
6
=
=
=
∩
=
Clearly
P(E ∩ F) = P(E) |
1 | 6932-6935 | Example 10 A die is thrown If E is the event ‘the number appearing is a multiple of
3’ and F be the event ‘the number appearing is even’ then find whether E and F are
independent Solution We know that the sample space is S = {1, 2, 3, 4, 5, 6}
Now
E = { 3, 6}, F = { 2, 4, 6} and E ∩ F = {6}
Then
P(E) = 2
1
3
1
1
, P(F)
and P(E
F)
6
3
6
2
6
=
=
=
∩
=
Clearly
P(E ∩ F) = P(E) P (F)
Hence
E and F are independent events |
1 | 6933-6936 | If E is the event ‘the number appearing is a multiple of
3’ and F be the event ‘the number appearing is even’ then find whether E and F are
independent Solution We know that the sample space is S = {1, 2, 3, 4, 5, 6}
Now
E = { 3, 6}, F = { 2, 4, 6} and E ∩ F = {6}
Then
P(E) = 2
1
3
1
1
, P(F)
and P(E
F)
6
3
6
2
6
=
=
=
∩
=
Clearly
P(E ∩ F) = P(E) P (F)
Hence
E and F are independent events Example 11 An unbiased die is thrown twice |
1 | 6934-6937 | Solution We know that the sample space is S = {1, 2, 3, 4, 5, 6}
Now
E = { 3, 6}, F = { 2, 4, 6} and E ∩ F = {6}
Then
P(E) = 2
1
3
1
1
, P(F)
and P(E
F)
6
3
6
2
6
=
=
=
∩
=
Clearly
P(E ∩ F) = P(E) P (F)
Hence
E and F are independent events Example 11 An unbiased die is thrown twice Let the event A be ‘odd number on the
first throw’ and B the event ‘odd number on the second throw’ |
1 | 6935-6938 | P (F)
Hence
E and F are independent events Example 11 An unbiased die is thrown twice Let the event A be ‘odd number on the
first throw’ and B the event ‘odd number on the second throw’ Check the independence
of the events A and B |
1 | 6936-6939 | Example 11 An unbiased die is thrown twice Let the event A be ‘odd number on the
first throw’ and B the event ‘odd number on the second throw’ Check the independence
of the events A and B Solution If all the 36 elementary events of the experiment are considered to be equally
likely, we have
P(A) = 18
1
36
=2
and
18
1
P(B)
36
2
Also
P(A ∩ B) = P (odd number on both throws)
= 9
1
36
4
=
Now
P(A) P(B) = 1
1
1
2
2
4
×
=
Clearly
P(A ∩ B) = P(A) × P(B)
Thus,
A and B are independent events
Example 12 Three coins are tossed simultaneously |
1 | 6937-6940 | Let the event A be ‘odd number on the
first throw’ and B the event ‘odd number on the second throw’ Check the independence
of the events A and B Solution If all the 36 elementary events of the experiment are considered to be equally
likely, we have
P(A) = 18
1
36
=2
and
18
1
P(B)
36
2
Also
P(A ∩ B) = P (odd number on both throws)
= 9
1
36
4
=
Now
P(A) P(B) = 1
1
1
2
2
4
×
=
Clearly
P(A ∩ B) = P(A) × P(B)
Thus,
A and B are independent events
Example 12 Three coins are tossed simultaneously Consider the event E ‘three heads
or three tails’, F ‘at least two heads’ and G ‘at most two heads’ |
1 | 6938-6941 | Check the independence
of the events A and B Solution If all the 36 elementary events of the experiment are considered to be equally
likely, we have
P(A) = 18
1
36
=2
and
18
1
P(B)
36
2
Also
P(A ∩ B) = P (odd number on both throws)
= 9
1
36
4
=
Now
P(A) P(B) = 1
1
1
2
2
4
×
=
Clearly
P(A ∩ B) = P(A) × P(B)
Thus,
A and B are independent events
Example 12 Three coins are tossed simultaneously Consider the event E ‘three heads
or three tails’, F ‘at least two heads’ and G ‘at most two heads’ Of the pairs (E,F),
(E,G) and (F,G), which are independent |
1 | 6939-6942 | Solution If all the 36 elementary events of the experiment are considered to be equally
likely, we have
P(A) = 18
1
36
=2
and
18
1
P(B)
36
2
Also
P(A ∩ B) = P (odd number on both throws)
= 9
1
36
4
=
Now
P(A) P(B) = 1
1
1
2
2
4
×
=
Clearly
P(A ∩ B) = P(A) × P(B)
Thus,
A and B are independent events
Example 12 Three coins are tossed simultaneously Consider the event E ‘three heads
or three tails’, F ‘at least two heads’ and G ‘at most two heads’ Of the pairs (E,F),
(E,G) and (F,G), which are independent which are dependent |
1 | 6940-6943 | Consider the event E ‘three heads
or three tails’, F ‘at least two heads’ and G ‘at most two heads’ Of the pairs (E,F),
(E,G) and (F,G), which are independent which are dependent Solution The sample space of the experiment is given by
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Clearly
E = {HHH, TTT}, F= {HHH, HHT, HTH, THH}
© NCERT
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PROBABILITY 545
and
G = {HHT, HTH, THH, HTT, THT, TTH, TTT}
Also
E ∩ F = {HHH}, E ∩ G = {TTT}, F ∩ G = { HHT, HTH, THH}
Therefore
P(E) = 2
1
4
1
7
, P(F)
, P(G)
8
4
8
2
8
=
=
=
=
and
P(E∩F) = 1
1
3
, P(E
G)
, P(F
G)
8
8
8
∩
=
∩
=
Also
P(E) |
1 | 6941-6944 | Of the pairs (E,F),
(E,G) and (F,G), which are independent which are dependent Solution The sample space of the experiment is given by
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Clearly
E = {HHH, TTT}, F= {HHH, HHT, HTH, THH}
© NCERT
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PROBABILITY 545
and
G = {HHT, HTH, THH, HTT, THT, TTH, TTT}
Also
E ∩ F = {HHH}, E ∩ G = {TTT}, F ∩ G = { HHT, HTH, THH}
Therefore
P(E) = 2
1
4
1
7
, P(F)
, P(G)
8
4
8
2
8
=
=
=
=
and
P(E∩F) = 1
1
3
, P(E
G)
, P(F
G)
8
8
8
∩
=
∩
=
Also
P(E) P(F) = 1
1
1
1
7
7
, P(E) P(G)
4
2
8
4
8
32
and
P(F) |
1 | 6942-6945 | which are dependent Solution The sample space of the experiment is given by
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Clearly
E = {HHH, TTT}, F= {HHH, HHT, HTH, THH}
© NCERT
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PROBABILITY 545
and
G = {HHT, HTH, THH, HTT, THT, TTH, TTT}
Also
E ∩ F = {HHH}, E ∩ G = {TTT}, F ∩ G = { HHT, HTH, THH}
Therefore
P(E) = 2
1
4
1
7
, P(F)
, P(G)
8
4
8
2
8
=
=
=
=
and
P(E∩F) = 1
1
3
, P(E
G)
, P(F
G)
8
8
8
∩
=
∩
=
Also
P(E) P(F) = 1
1
1
1
7
7
, P(E) P(G)
4
2
8
4
8
32
and
P(F) P(G) = 1
7
7
2
8
16
Thus
P(E ∩ F) = P(E) |
1 | 6943-6946 | Solution The sample space of the experiment is given by
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Clearly
E = {HHH, TTT}, F= {HHH, HHT, HTH, THH}
© NCERT
not to be republished
PROBABILITY 545
and
G = {HHT, HTH, THH, HTT, THT, TTH, TTT}
Also
E ∩ F = {HHH}, E ∩ G = {TTT}, F ∩ G = { HHT, HTH, THH}
Therefore
P(E) = 2
1
4
1
7
, P(F)
, P(G)
8
4
8
2
8
=
=
=
=
and
P(E∩F) = 1
1
3
, P(E
G)
, P(F
G)
8
8
8
∩
=
∩
=
Also
P(E) P(F) = 1
1
1
1
7
7
, P(E) P(G)
4
2
8
4
8
32
and
P(F) P(G) = 1
7
7
2
8
16
Thus
P(E ∩ F) = P(E) P(F)
P(E ∩ G) ≠ P(E) |
1 | 6944-6947 | P(F) = 1
1
1
1
7
7
, P(E) P(G)
4
2
8
4
8
32
and
P(F) P(G) = 1
7
7
2
8
16
Thus
P(E ∩ F) = P(E) P(F)
P(E ∩ G) ≠ P(E) P(G)
and
P(F ∩ G) ≠ P (F) |
1 | 6945-6948 | P(G) = 1
7
7
2
8
16
Thus
P(E ∩ F) = P(E) P(F)
P(E ∩ G) ≠ P(E) P(G)
and
P(F ∩ G) ≠ P (F) P(G)
Hence, the events (E and F) are independent, and the events (E and G) and
(F and G) are dependent |
1 | 6946-6949 | P(F)
P(E ∩ G) ≠ P(E) P(G)
and
P(F ∩ G) ≠ P (F) P(G)
Hence, the events (E and F) are independent, and the events (E and G) and
(F and G) are dependent Example 13 Prove that if E and F are independent events, then so are the events
E and F′ |
1 | 6947-6950 | P(G)
and
P(F ∩ G) ≠ P (F) P(G)
Hence, the events (E and F) are independent, and the events (E and G) and
(F and G) are dependent Example 13 Prove that if E and F are independent events, then so are the events
E and F′ Solution Since E and F are independent, we have
P(E ∩ F) = P(E) |
1 | 6948-6951 | P(G)
Hence, the events (E and F) are independent, and the events (E and G) and
(F and G) are dependent Example 13 Prove that if E and F are independent events, then so are the events
E and F′ Solution Since E and F are independent, we have
P(E ∩ F) = P(E) P(F) |
1 | 6949-6952 | Example 13 Prove that if E and F are independent events, then so are the events
E and F′ Solution Since E and F are independent, we have
P(E ∩ F) = P(E) P(F) (1)
From the venn diagram in Fig 13 |
1 | 6950-6953 | Solution Since E and F are independent, we have
P(E ∩ F) = P(E) P(F) (1)
From the venn diagram in Fig 13 3, it is clear
that E ∩ F and E ∩ F′ are mutually exclusive events
and also E =(E ∩ F) ∪ (E ∩ F′) |
1 | 6951-6954 | P(F) (1)
From the venn diagram in Fig 13 3, it is clear
that E ∩ F and E ∩ F′ are mutually exclusive events
and also E =(E ∩ F) ∪ (E ∩ F′) Therefore
P(E) = P(E ∩ F) + P(E ∩ F′)
or
P(E ∩ F′) = P(E) − P(E ∩ F)
= P(E) − P(E) |
1 | 6952-6955 | (1)
From the venn diagram in Fig 13 3, it is clear
that E ∩ F and E ∩ F′ are mutually exclusive events
and also E =(E ∩ F) ∪ (E ∩ F′) Therefore
P(E) = P(E ∩ F) + P(E ∩ F′)
or
P(E ∩ F′) = P(E) − P(E ∩ F)
= P(E) − P(E) P(F)
(by (1))
= P(E) (1−P(F))
= P(E) |
1 | 6953-6956 | 3, it is clear
that E ∩ F and E ∩ F′ are mutually exclusive events
and also E =(E ∩ F) ∪ (E ∩ F′) Therefore
P(E) = P(E ∩ F) + P(E ∩ F′)
or
P(E ∩ F′) = P(E) − P(E ∩ F)
= P(E) − P(E) P(F)
(by (1))
= P(E) (1−P(F))
= P(E) P(F′)
Hence, E and F′ are independent
(E
∩F )’
(E
F)
’∩
E
F
S
(E
F)
∩
(E
F )
’
’
∩
Fig 13 |
1 | 6954-6957 | Therefore
P(E) = P(E ∩ F) + P(E ∩ F′)
or
P(E ∩ F′) = P(E) − P(E ∩ F)
= P(E) − P(E) P(F)
(by (1))
= P(E) (1−P(F))
= P(E) P(F′)
Hence, E and F′ are independent
(E
∩F )’
(E
F)
’∩
E
F
S
(E
F)
∩
(E
F )
’
’
∩
Fig 13 3
© NCERT
not to be republished
546
MATHEMATICS
�Note In a similar manner, it can be shown that if the events E and F are
independent, then
(a)
E′ and F are independent,
(b)
E′ and F′ are independent
Example 14 If A and B are two independent events, then the probability of occurrence
of at least one of A and B is given by 1– P(A′) P(B′)
Solution We have
P(at least one of A and B) = P(A ∪ B)
= P(A) + P(B) − P(A ∩ B)
= P(A) + P(B) − P(A) P(B)
= P(A) + P(B) [1−P(A)]
= P(A) + P(B) |
1 | 6955-6958 | P(F)
(by (1))
= P(E) (1−P(F))
= P(E) P(F′)
Hence, E and F′ are independent
(E
∩F )’
(E
F)
’∩
E
F
S
(E
F)
∩
(E
F )
’
’
∩
Fig 13 3
© NCERT
not to be republished
546
MATHEMATICS
�Note In a similar manner, it can be shown that if the events E and F are
independent, then
(a)
E′ and F are independent,
(b)
E′ and F′ are independent
Example 14 If A and B are two independent events, then the probability of occurrence
of at least one of A and B is given by 1– P(A′) P(B′)
Solution We have
P(at least one of A and B) = P(A ∪ B)
= P(A) + P(B) − P(A ∩ B)
= P(A) + P(B) − P(A) P(B)
= P(A) + P(B) [1−P(A)]
= P(A) + P(B) P(A′)
= 1− P(A′) + P(B) P(A′)
= 1− P(A′) [1− P(B)]
= 1− P(A′) P (B′)
EXERCISE 13 |
1 | 6956-6959 | P(F′)
Hence, E and F′ are independent
(E
∩F )’
(E
F)
’∩
E
F
S
(E
F)
∩
(E
F )
’
’
∩
Fig 13 3
© NCERT
not to be republished
546
MATHEMATICS
�Note In a similar manner, it can be shown that if the events E and F are
independent, then
(a)
E′ and F are independent,
(b)
E′ and F′ are independent
Example 14 If A and B are two independent events, then the probability of occurrence
of at least one of A and B is given by 1– P(A′) P(B′)
Solution We have
P(at least one of A and B) = P(A ∪ B)
= P(A) + P(B) − P(A ∩ B)
= P(A) + P(B) − P(A) P(B)
= P(A) + P(B) [1−P(A)]
= P(A) + P(B) P(A′)
= 1− P(A′) + P(B) P(A′)
= 1− P(A′) [1− P(B)]
= 1− P(A′) P (B′)
EXERCISE 13 2
1 |
1 | 6957-6960 | 3
© NCERT
not to be republished
546
MATHEMATICS
�Note In a similar manner, it can be shown that if the events E and F are
independent, then
(a)
E′ and F are independent,
(b)
E′ and F′ are independent
Example 14 If A and B are two independent events, then the probability of occurrence
of at least one of A and B is given by 1– P(A′) P(B′)
Solution We have
P(at least one of A and B) = P(A ∪ B)
= P(A) + P(B) − P(A ∩ B)
= P(A) + P(B) − P(A) P(B)
= P(A) + P(B) [1−P(A)]
= P(A) + P(B) P(A′)
= 1− P(A′) + P(B) P(A′)
= 1− P(A′) [1− P(B)]
= 1− P(A′) P (B′)
EXERCISE 13 2
1 If P(A)
53
and P (B)
51
, find P (A ∩ B) if A and B are independent events |
1 | 6958-6961 | P(A′)
= 1− P(A′) + P(B) P(A′)
= 1− P(A′) [1− P(B)]
= 1− P(A′) P (B′)
EXERCISE 13 2
1 If P(A)
53
and P (B)
51
, find P (A ∩ B) if A and B are independent events 2 |
1 | 6959-6962 | 2
1 If P(A)
53
and P (B)
51
, find P (A ∩ B) if A and B are independent events 2 Two cards are drawn at random and without replacement from a pack of 52
playing cards |
1 | 6960-6963 | If P(A)
53
and P (B)
51
, find P (A ∩ B) if A and B are independent events 2 Two cards are drawn at random and without replacement from a pack of 52
playing cards Find the probability that both the cards are black |
1 | 6961-6964 | 2 Two cards are drawn at random and without replacement from a pack of 52
playing cards Find the probability that both the cards are black 3 |
1 | 6962-6965 | Two cards are drawn at random and without replacement from a pack of 52
playing cards Find the probability that both the cards are black 3 A box of oranges is inspected by examining three randomly selected oranges
drawn without replacement |
1 | 6963-6966 | Find the probability that both the cards are black 3 A box of oranges is inspected by examining three randomly selected oranges
drawn without replacement If all the three oranges are good, the box is approved
for sale, otherwise, it is rejected |
1 | 6964-6967 | 3 A box of oranges is inspected by examining three randomly selected oranges
drawn without replacement If all the three oranges are good, the box is approved
for sale, otherwise, it is rejected Find the probability that a box containing 15
oranges out of which 12 are good and 3 are bad ones will be approved for sale |
1 | 6965-6968 | A box of oranges is inspected by examining three randomly selected oranges
drawn without replacement If all the three oranges are good, the box is approved
for sale, otherwise, it is rejected Find the probability that a box containing 15
oranges out of which 12 are good and 3 are bad ones will be approved for sale 4 |
1 | 6966-6969 | If all the three oranges are good, the box is approved
for sale, otherwise, it is rejected Find the probability that a box containing 15
oranges out of which 12 are good and 3 are bad ones will be approved for sale 4 A fair coin and an unbiased die are tossed |
1 | 6967-6970 | Find the probability that a box containing 15
oranges out of which 12 are good and 3 are bad ones will be approved for sale 4 A fair coin and an unbiased die are tossed Let A be the event ‘head appears on
the coin’ and B be the event ‘3 on the die’ |
1 | 6968-6971 | 4 A fair coin and an unbiased die are tossed Let A be the event ‘head appears on
the coin’ and B be the event ‘3 on the die’ Check whether A and B are
independent events or not |
1 | 6969-6972 | A fair coin and an unbiased die are tossed Let A be the event ‘head appears on
the coin’ and B be the event ‘3 on the die’ Check whether A and B are
independent events or not 5 |
1 | 6970-6973 | Let A be the event ‘head appears on
the coin’ and B be the event ‘3 on the die’ Check whether A and B are
independent events or not 5 A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed |
1 | 6971-6974 | Check whether A and B are
independent events or not 5 A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed Let A be the event,
‘the number is even,’ and B be the event, ‘the number is red’ |
1 | 6972-6975 | 5 A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed Let A be the event,
‘the number is even,’ and B be the event, ‘the number is red’ Are A and B
independent |
1 | 6973-6976 | A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed Let A be the event,
‘the number is even,’ and B be the event, ‘the number is red’ Are A and B
independent 6 |
1 | 6974-6977 | Let A be the event,
‘the number is even,’ and B be the event, ‘the number is red’ Are A and B
independent 6 Let E and F be events with P(E)
53
, P(F)
=103
and P (E ∩ F) = 1
5 |
1 | 6975-6978 | Are A and B
independent 6 Let E and F be events with P(E)
53
, P(F)
=103
and P (E ∩ F) = 1
5 Are
E and F independent |
1 | 6976-6979 | 6 Let E and F be events with P(E)
53
, P(F)
=103
and P (E ∩ F) = 1
5 Are
E and F independent © NCERT
not to be republished
PROBABILITY 547
7 |
1 | 6977-6980 | Let E and F be events with P(E)
53
, P(F)
=103
and P (E ∩ F) = 1
5 Are
E and F independent © NCERT
not to be republished
PROBABILITY 547
7 Given that the events A and B are such that P(A) = 1
2 , P(A ∪ B) = 3
5 and
P(B) = p |
1 | 6978-6981 | Are
E and F independent © NCERT
not to be republished
PROBABILITY 547
7 Given that the events A and B are such that P(A) = 1
2 , P(A ∪ B) = 3
5 and
P(B) = p Find p if they are (i) mutually exclusive (ii) independent |
1 | 6979-6982 | © NCERT
not to be republished
PROBABILITY 547
7 Given that the events A and B are such that P(A) = 1
2 , P(A ∪ B) = 3
5 and
P(B) = p Find p if they are (i) mutually exclusive (ii) independent 8 |
1 | 6980-6983 | Given that the events A and B are such that P(A) = 1
2 , P(A ∪ B) = 3
5 and
P(B) = p Find p if they are (i) mutually exclusive (ii) independent 8 Let A and B be independent events with P(A) = 0 |
1 | 6981-6984 | Find p if they are (i) mutually exclusive (ii) independent 8 Let A and B be independent events with P(A) = 0 3 and P(B) = 0 |
1 | 6982-6985 | 8 Let A and B be independent events with P(A) = 0 3 and P(B) = 0 4 |
1 | 6983-6986 | Let A and B be independent events with P(A) = 0 3 and P(B) = 0 4 Find
(i) P(A ∩ B)
(ii) P(A ∪ B)
(iii) P(A|B)
(iv) P(B|A)
9 |
1 | 6984-6987 | 3 and P(B) = 0 4 Find
(i) P(A ∩ B)
(ii) P(A ∪ B)
(iii) P(A|B)
(iv) P(B|A)
9 If A and B are two events such that P(A) = 1
4 , P (B) = 1
2 and P(A ∩ B) = 1
8 ,
find P (not A and not B) |
1 | 6985-6988 | 4 Find
(i) P(A ∩ B)
(ii) P(A ∪ B)
(iii) P(A|B)
(iv) P(B|A)
9 If A and B are two events such that P(A) = 1
4 , P (B) = 1
2 and P(A ∩ B) = 1
8 ,
find P (not A and not B) 10 |
1 | 6986-6989 | Find
(i) P(A ∩ B)
(ii) P(A ∪ B)
(iii) P(A|B)
(iv) P(B|A)
9 If A and B are two events such that P(A) = 1
4 , P (B) = 1
2 and P(A ∩ B) = 1
8 ,
find P (not A and not B) 10 Events A and B are such that P (A) = 1
2 , P(B) = 7
12 and P(not A or not B) = 1
4 |
1 | 6987-6990 | If A and B are two events such that P(A) = 1
4 , P (B) = 1
2 and P(A ∩ B) = 1
8 ,
find P (not A and not B) 10 Events A and B are such that P (A) = 1
2 , P(B) = 7
12 and P(not A or not B) = 1
4 State whether A and B are independent |
1 | 6988-6991 | 10 Events A and B are such that P (A) = 1
2 , P(B) = 7
12 and P(not A or not B) = 1
4 State whether A and B are independent 11 |
1 | 6989-6992 | Events A and B are such that P (A) = 1
2 , P(B) = 7
12 and P(not A or not B) = 1
4 State whether A and B are independent 11 Given two independent events A and B such that P(A) = 0 |
1 | 6990-6993 | State whether A and B are independent 11 Given two independent events A and B such that P(A) = 0 3, P(B) = 0 |
1 | 6991-6994 | 11 Given two independent events A and B such that P(A) = 0 3, P(B) = 0 6 |
1 | 6992-6995 | Given two independent events A and B such that P(A) = 0 3, P(B) = 0 6 Find
(i) P(A and B)
(ii) P(A and not B)
(iii) P(A or B)
(iv) P(neither A nor B)
12 |
1 | 6993-6996 | 3, P(B) = 0 6 Find
(i) P(A and B)
(ii) P(A and not B)
(iii) P(A or B)
(iv) P(neither A nor B)
12 A die is tossed thrice |
1 | 6994-6997 | 6 Find
(i) P(A and B)
(ii) P(A and not B)
(iii) P(A or B)
(iv) P(neither A nor B)
12 A die is tossed thrice Find the probability of getting an odd number at least once |
1 | 6995-6998 | Find
(i) P(A and B)
(ii) P(A and not B)
(iii) P(A or B)
(iv) P(neither A nor B)
12 A die is tossed thrice Find the probability of getting an odd number at least once 13 |
1 | 6996-6999 | A die is tossed thrice Find the probability of getting an odd number at least once 13 Two balls are drawn at random with replacement from a box containing 10 black
and 8 red balls |
1 | 6997-7000 | Find the probability of getting an odd number at least once 13 Two balls are drawn at random with replacement from a box containing 10 black
and 8 red balls Find the probability that
(i) both balls are red |
1 | 6998-7001 | 13 Two balls are drawn at random with replacement from a box containing 10 black
and 8 red balls Find the probability that
(i) both balls are red (ii) first ball is black and second is red |
1 | 6999-7002 | Two balls are drawn at random with replacement from a box containing 10 black
and 8 red balls Find the probability that
(i) both balls are red (ii) first ball is black and second is red (iii) one of them is black and other is red |
1 | 7000-7003 | Find the probability that
(i) both balls are red (ii) first ball is black and second is red (iii) one of them is black and other is red 14 |
1 | 7001-7004 | (ii) first ball is black and second is red (iii) one of them is black and other is red 14 Probability of solving specific problem independently by A and B are 1
2 and 1
3
respectively |
1 | 7002-7005 | (iii) one of them is black and other is red 14 Probability of solving specific problem independently by A and B are 1
2 and 1
3
respectively If both try to solve the problem independently, find the probability
that
(i) the problem is solved
(ii) exactly one of them solves the problem |
1 | 7003-7006 | 14 Probability of solving specific problem independently by A and B are 1
2 and 1
3
respectively If both try to solve the problem independently, find the probability
that
(i) the problem is solved
(ii) exactly one of them solves the problem 15 |
1 | 7004-7007 | Probability of solving specific problem independently by A and B are 1
2 and 1
3
respectively If both try to solve the problem independently, find the probability
that
(i) the problem is solved
(ii) exactly one of them solves the problem 15 One card is drawn at random from a well shuffled deck of 52 cards |
1 | 7005-7008 | If both try to solve the problem independently, find the probability
that
(i) the problem is solved
(ii) exactly one of them solves the problem 15 One card is drawn at random from a well shuffled deck of 52 cards In which of
the following cases are the events E and F independent |
1 | 7006-7009 | 15 One card is drawn at random from a well shuffled deck of 52 cards In which of
the following cases are the events E and F independent (i) E : ‘the card drawn is a spade’
F : ‘the card drawn is an ace’
(ii) E : ‘the card drawn is black’
F : ‘the card drawn is a king’
(iii) E : ‘the card drawn is a king or queen’
F : ‘the card drawn is a queen or jack’ |
1 | 7007-7010 | One card is drawn at random from a well shuffled deck of 52 cards In which of
the following cases are the events E and F independent (i) E : ‘the card drawn is a spade’
F : ‘the card drawn is an ace’
(ii) E : ‘the card drawn is black’
F : ‘the card drawn is a king’
(iii) E : ‘the card drawn is a king or queen’
F : ‘the card drawn is a queen or jack’ © NCERT
not to be republished
548
MATHEMATICS
16 |
1 | 7008-7011 | In which of
the following cases are the events E and F independent (i) E : ‘the card drawn is a spade’
F : ‘the card drawn is an ace’
(ii) E : ‘the card drawn is black’
F : ‘the card drawn is a king’
(iii) E : ‘the card drawn is a king or queen’
F : ‘the card drawn is a queen or jack’ © NCERT
not to be republished
548
MATHEMATICS
16 In a hostel, 60% of the students read Hindi news paper, 40% read English news
paper and 20% read both Hindi and English news papers |
1 | 7009-7012 | (i) E : ‘the card drawn is a spade’
F : ‘the card drawn is an ace’
(ii) E : ‘the card drawn is black’
F : ‘the card drawn is a king’
(iii) E : ‘the card drawn is a king or queen’
F : ‘the card drawn is a queen or jack’ © NCERT
not to be republished
548
MATHEMATICS
16 In a hostel, 60% of the students read Hindi news paper, 40% read English news
paper and 20% read both Hindi and English news papers A student is selected
at random |
1 | 7010-7013 | © NCERT
not to be republished
548
MATHEMATICS
16 In a hostel, 60% of the students read Hindi news paper, 40% read English news
paper and 20% read both Hindi and English news papers A student is selected
at random (a) Find the probability that she reads neither Hindi nor English news papers |
1 | 7011-7014 | In a hostel, 60% of the students read Hindi news paper, 40% read English news
paper and 20% read both Hindi and English news papers A student is selected
at random (a) Find the probability that she reads neither Hindi nor English news papers (b) If she reads Hindi news paper, find the probability that she reads English
news paper |
1 | 7012-7015 | A student is selected
at random (a) Find the probability that she reads neither Hindi nor English news papers (b) If she reads Hindi news paper, find the probability that she reads English
news paper (c) If she reads English news paper, find the probability that she reads Hindi
news paper |
1 | 7013-7016 | (a) Find the probability that she reads neither Hindi nor English news papers (b) If she reads Hindi news paper, find the probability that she reads English
news paper (c) If she reads English news paper, find the probability that she reads Hindi
news paper Choose the correct answer in Exercises 17 and 18 |
1 | 7014-7017 | (b) If she reads Hindi news paper, find the probability that she reads English
news paper (c) If she reads English news paper, find the probability that she reads Hindi
news paper Choose the correct answer in Exercises 17 and 18 17 |
1 | 7015-7018 | (c) If she reads English news paper, find the probability that she reads Hindi
news paper Choose the correct answer in Exercises 17 and 18 17 The probability of obtaining an even prime number on each die, when a pair of
dice is rolled is
(A) 0
(B) 1
3
(C)
121
(D)
1
36
18 |
1 | 7016-7019 | Choose the correct answer in Exercises 17 and 18 17 The probability of obtaining an even prime number on each die, when a pair of
dice is rolled is
(A) 0
(B) 1
3
(C)
121
(D)
1
36
18 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
13 |
1 | 7017-7020 | 17 The probability of obtaining an even prime number on each die, when a pair of
dice is rolled is
(A) 0
(B) 1
3
(C)
121
(D)
1
36
18 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
13 5 Bayes' Theorem
Consider that there are two bags I and II |
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